A  TEXT-BOOK 


ON  THE 


METHOD  OF  LEAST  SQUARES. 


WORKS  OF 
THE  LATE  MANSFIELD  MERRIMAN 

PUBLISHED    BY 

JOHN  WILEY  &  SONS,  Inc. 

Mechanics  of  Materials,  64th  Thousand,  si  by  8 J. 
Treatise  on  Hydraulics,  56th  Thousand,  si  by  8^. 
Sanitary  Engineering,  6th  Thousand,  6  by  9. 
Method  of  Least  Squares,  loth  Thousand,  6  by  9. 
Elements  of  Mechanics,  i6th  Thousand,  s  by  7^. 
Elements  of  Hydraulics,  8th  Thousand,  s  by  73. 
Mathematical  Tables  for  Class  Use,  5  by  7j. 
Solution  of  Equations,  6  by  9. 

revised  by 
THADDEUS  MERRIMAN 

Strength  of  Materials,  S4th  Thousand,  S  by  7^. 

By  MANSFIELD  MERRIMAN  and  HENRY  S.  JACOBY 

TEXT-BOOK  ON  ROOFS  AND  BRIDGES 

Part  I.      Stresses,  28th  Thousand,  6  by  9. 
Part  II.     Graphic  Statics,  26th  Thousand,  6  by  9. 
Part  III.  Bridge  Design.     New  Edition  in  Preparation. 
Part  IV.    Higher  Structures,  nth  Thousand,  6  by  9. 

By   MANSFIELD   MERRIMAN  and  JOHN  P.  BROOKS 

Handbook  for  Surveyors,  loth  Thousand,  4  by  6\. 

MANSFIELD  MERRIMAN,  Editor-in-Chief 

American     Civil    Engineers'     Handbook,     soth     Thousand, 
44  by  7.     Genuine  leather  or  flexible  binding. 

Edited  by 
MANSFIELD    MERRIMAN   and  ROBERT   S.   WOODWARD 

MATHEMATICAL     MONOGRAPHS,     Twenty     Volumes,     6 
by  9. 


s        1  «  a 


A  TEXT-BOOiCriV 


ON  THE 


Method  of  Least  Squares. 


BY 

MANSFIELD    MERRIMAN, 

LATE   MEMBER   OF    AMERICAN   MATHEMATICAL    SOCIETY 


EIGHTH  EDITION,  REVISED 
TOTAL   ISSUE,    ELEVEN   THOUSAND. 


NEW  YORK 

JOHN  WILEY  &  SONS,  Inc. 
London:    CHAPMAN  &  HALL.   Limited 


Ml 


I 


ASTRONOMY   DEPT. 

Copyright,  1884, 
BY  MANSFIELD  MERRIMAN. 

Copyright  Renewed,  191  i, 
BY  MANSFIELD  MERRIMAN, 


Printed  in  U.  S.  A, 

6/29 


PRESS    OF 

BRAUNWOHTH    &    CO..    INC. 

BOOK    MANUFACTURERS 

BROOKLYN,   NEWYORK 


Y 


PREFACE  TO  THE  FIRST  EDITION. 


The  "Elements  of  the  Method  of  Least  Squares,"  published 
in  1877,  was  written  with  two  objects  in  view  :  first,  to  present 
the  fundamental  principles  and  processes  of  the  subject  in  so 
plain  a  manner,  and  to  illustrate  their  application  by  such 
simple  and  practical  examples,  as  to  render  it  accessible  to 
civil  engineers  who  have  not  had  the  benefit  of  extended 
mathematical  training;  and,  secondly,  to  give  an  elementary 
exposition  of  the  theory  which  would  be  adapted  to  the  needs 
of  a  large  and  constantly  increasing  class  of  students. 

In  preparing  the  following  pages  the  author  has  likewise 
kept  these  two  objects  continually  in  mind.  While  the  for- 
mer work  has  been  used  as  a  basis,  the  alterations  and 
additions  have  been  so  numerous  and  radical  as  to  render 
this  a  new  and  distinct  book  rather  than  a  second  edition. 
The  arrangement  of  the  theoretical  and  practical  parts  is 
entirely  different.  In  Chapters  I  to  IV  is  presented  the 
mathematical  development  of  the  principles,  methods,  and 
formulas;    while  in  Chapters  V  to  IX  the  application  of  these 


r»V| 


67874 


IV  PREFACE. 

to  the  different  classes  of  observations  is  made,  and  illus- 
trated by  numerous  practical  examples.  For  the  use  of  both 
students  and  engineers,  it  is  believed  that  this  plan  will 
prove  more  advantageous  than  the  one  previously  followed. 
Hagen's  deduction  of  the  law  of  probability  of  error  is  given, 
as  well  as  that  of  Gauss.  More  attention  is  paid  to  the 
laws  of  the  propagation  of  error,  the  solution  of  normal  equa- 
tions, and  the  deduction  of  empirical  formulas.  Many  new 
illustrative  examples  of  the  adjustment  and  comparison  of 
observations  have  been  selected  from  actual  practice,  and 
are  discussed  m  detail.  At  the  end  of  each  chapter  are 
given  a  few  problems  or  queries  ;  and  in  the  Appendix  are 
eight  tables  for  abridging  computations. 

MANSFIELD    MERRIMAN. 

NOTE   TO   THE    EIGHTH    EDITION. 

The  seventh  edition  was  the  result  of  a  thorough  revision 
and  was  enlarged  by  the  addition  of  new  matter  on  the  solu- 
tion of  normal  equations,  on  the  uncertainty  of  the  probable 
error,  and  on  the  median.  In  this  edition  all  known  eirors 
have  been  corrected  and  an  alphabetical  index  has  been 
added.  M.  M. 


CONTENTS. 


CHAPTER    I. 


/ 


INTRODUCTION. 


Classification  of  Observations 
Errors  of  Observations  . 
Principles  of  Probability   . 
Problems       


3 
6 

12 


CHAPTER    II. 


LAW   OF  PROBABILITY   OF  ERROR. 


Axioms  derived  from  Experience     . 
The  Probability  Curve    .... 
First  Deduction  of  the  Law  of  Error  . 
Second  Deduction  of  the  Law  of  Error 
Discussion  of  the  Probability  Curve     . 
The  Probability  Integral 
Comparison  of  Theory  and  Experience  . 
Remarks  on  the  Fundamental  Formulas 
Problems  and  Queries 


IS 

I? 

22 

25 
27 

33 
35 


CHAPTER   III. 


THE   ADJUSTMENT   OF   OBSERVATIONS. 

Weights  of  Observations '        •  3" 

The  Principle  of  Least  Squares       .        =. 3° 

Direct  Observations  on  a  Single  Quantity      .        .        .        .        •  4^ 

Independent  Observations  of  Equal  Weight       .        .        .        o        .  43 

Independent  Observations  of  Unequal  Weight      ....  51 

Solution  of  Normal  Equations 56 

Conditioned  Observations'' 57 

Problems ^ ^5 

V 


Vi  CONTENTS. 

CHAPTER    IV.   ^ 
THE   PRECISION   OF    OBSERVATIONS 

The  Probable  Error .        ,  gg 

Prolable  Error  of  the  Arithmetical  Mean                                         ,.  jo 
Probable  Error  of  the  General  Mean       =        ,                                .72 

Laws  of  Propagation  of  Error         •..-.,..  75 

Probable  Errors  for  Independent  Observations    .        ,        ..        ,  79 

Probable  Errors  for  Conditioned  Observations        -        ...  86 

Problems      .=.....,..,,.  87 


CHAPTER   V,   ■' 

DIRECT   OBSERVATIONS    ON  A    SINGLE   QUANTIIY. 

Observations  of  Equal  Weight        „.».        =        ,.,  88 

Shorter  Formulas  for  Probable  Error      =       =        =,..  92 

Observations  of  Unequal  Weight  .        .        »        .        .        e        ,  95 

Problems ,       =       .  99 


CHAPTER  VL    ^ 

FUNCTIONS    OF   OBSERVED    QUANTITIES. 

Linear  Measurements  .........        c  .      toi 

Angle  Measurements =  104 

Precision  of  Areas        .        .        .        , ,=  .      106 

Remarks  and  Problems 107 


CHAPTER   VII. 

INDEPENDENT   OBSERVATIONS    ON  SEVERAL    QUANTITIES, 

Method  of  Procedure  .......,„,  X09 

Discussion  of  Level  Lines      ......,        =        ,  no 

Angles  at  a  Station     ........        t        ..  117 

Empirical  Constants         .        .        .        .        .        .       ,        .        »        ,  124 

Empirical  Formulas      ....,>....  130 

Problems      ........  139 


CONTENTS.  VI 1 

CHAPTER   VIII. 
CONDITIONED    OBSERVATIONS. 

The  Two  Methods  of  Procedure 141 

Angles  of  a  Triangle 142 

Angles  at  a  Station i45 

Angles  of  a  Quadrilateral i47 

Simple  Triangulation 152 

Levelling '54 

Problems '^ 

CHAPTER   IX. 

THE    DISCUSSION    OF   OBSERVATIONS. 

Probability  of  Errors 162 

The  Rejection  of  Doubtful  Observations 166 

Constant  Errors 169 

Social  Statistics i?^ 

Problems <■  *74 

CHAPTER   X. 
SOLUTION  OF  NORMAL   EQUATIONS. 

Three  Normal  Equations t  I75 

Formation  of  Normal  Equations J 77 

Gauss's  Method  of  Solution 181 

Weighted  Observations •        •        .        .  187 

Logarithmic  Computations  ....•.*••  190 

Probable  Errors  of  Adjusted  Values ^95 

Problems ••  ^98 


CHAPTER  XL 

appendix  and  tables. 

Observations  Involving  Non-Linear  Equations    .        .        •        •  200 

Mean  and  Probable  Error •        «        •  204 

Uncertainty  of  the  Probable  Error     ....<>  206 

The  Median 208 


VI 11  CONTENTS. 

History  and  Literature ,        .        ,        ,  211 

Constant  Numbers 214 

Answers  to  Problems  ;  and  Notes 215 

Description  of  the  Tables 219 

TABLES. 

I.  Values  of  the  Probability  Integral  for  Argument  hx       .  220 

II.  Values  of  the  Probability  Integral  for  Argument  —        .  221 

r 

III.  For  Computing  Probable  Errors  by  Formulas  (20)  and  (21)  222 

IV.  For  Computing  Probable  Errors  by  Formulas  (35)  and  (36)  223 
V.  Common  Logarithms  of  Numbers   ......  224 

VI.  Squares  of  Numbers 226 

VII.   For  Applying  Chauvenet's  Criterion 228 

VIII.  Squares  of  Rpxiprocals 228 


INDEX 229 


a    >  ••• 


A  TEXT-BOOK 


ON   THE 


METHOD    OF    LEAST   SQUARES. 


CHAPTER   I. 

INTRODUCTION. 


I.  The  Method  of  Least  Squares  has  for  its  object  the 
adjustment  and  comparison  of  observations.  The  adjustment 
of  observations  is  rendered  necessary  by  the  fact,  that  when 
several  precise  measurements  are  made,  even  upon  the  same 
quantity  under  apparently  similar  conditions,  the  results  do  not 
agree.  The  absolutely  true  values  of  the  observed  quantities 
cannot  in  general  be  found,  but  instead  must  be  accepted  and 
used  values,  derived  from  the  combination  and  adjustment  of 
the  measurements,  which  are  the  most  probable,  and  hence  the 
best.  The  comparison  of  observations  is  necessary  in  order  to 
determine  the  relative  degrees  of  precision  of  different  sets 
of  measurements  made  under  different  circumstances,  either 
for  the  purpose  of  properly  combining  and  adjusting  them,  or 
to  ascertain  the  best  methods  of  observation. 


2  INTRODUCTION.  I. 

-    ■  "  Classification  of  Observations. 

2.  Direct  observations  are  those  which  are  made  directly 
upon  the  quantity  whose  magnitude  is  to  be  determined.  Such 
are  measurements  of  a  line  by  direct  chaining,  or  measurements 
of  an  angle  by  direct  reading  with  a  transit.  They  occur  in  the 
daily  practice  of  every  engineer. 

Indirect  observations  are  not  made  upon  the  quantity  whose 
size  is  to  be  measured,  but  upon  some  other  quantity  or  quanti- 
ties related  to  it.  Such  are  measurements  of  a  line  through 
a  triangulation  by  means  of  a  base  and  observed  angles,  meas- 
urements of  an  angle  by  regarding  it  as  the  sum  or  difference 
of  other  angles,  the  determination  of  the  difference  of  level  of 
points  by  readings  upon  graduated  rods  set  up  at  different 
places,  the  determination  of  latitude  by  observing  the  altitude 
of  stars,  etc.  In  fact,  the  majority  of  observations  in  engineer- 
ing and  physical  science  generally  belong  to  this  class. 

3.  Conditioned  observations  may  be  either  direct  or  indirect, 
but  are  subject  to  some  rigorous  requirement  or  condition  im- 
posed in  advance  from  theoretical  considerations.  As  such 
may  be  mentioned  :  the  three  measured  angles  in  a  plane  tri- 
angle must  be  so  adjusted  that  their  sum  shall  be  exactly  180°  ; 
the  sum  of  all  the  percentages  in  a  chemical  analysis  must 
equal  100;  and  the  sum  of  the  northings  must  equal  the  sum 
of  the  southings  in  any  traverse  which  begins  and  ends  at  the 
same  point. 

Independent  observations  may  be  either  direct  or  indirect, 
but  are  subject  to  no  rigorous  conditions.  Measurements  on 
two  of  the  angles  of  a  triangle,  for  instance,  are  independent  ; 
for  the  observed  quantities  can  have  no  necessary  geometrical 
dependence  one  upon  the  other. 

4.  As  an  illustration  of  these  classes,  consider  the  angles 


5- 


ERRORS  OF  OBSERVATIONS. 


,M 


AOB  and  BOC,  having  their  vertices  at  the  same  point,  O 
(Fig.  i).  If  a  transit  or  theodolite  be  set  at  O,  and  the  angle 
AOB  or  BOC  be  measured,  each  of  these  measurements  is 
a  direct  observation.  If,  however,  an  au.xiliary  station  AT  be 
established,  and  the  angles  MOA,  MOB,  and  MOC  be  read, 
the  observations  on  AOB  and  BOC  are  indirect.  Moreover, 
whether  observed  di- 
rectly or  indirectly,  the 
values  obtained  for 
AOB  and  BOC  are 
independent  of  each 
other.  But  if  the  three 
angles  AOB,  BOC,  and 
AOC  be  measured, 
these  observations  are 
conditioned,  or  subject 
to  the  rigorous  geomet- 
rical requirement,  that, 
when  finally  adjusted,  AOB  plus  BOC  must  equal  AOC ;  and 
no  system  of  values  can  be  adopted  for  these  three  angles  which 
does  not  exactly  satisfy  this  condition. 

Again  :  if  the  sides  and  angles  of  a  field  are  measured,  each 
observation  taken  alone  is  direct.  If  its  area  is  found  from 
the  sides  and  angles,  the  measurement  of  that  area  is  indirect. 
Further  :  any  two  sides  considered  are  independent  of  each 
other ;  but,  if  all  the  sides  and  angles  be  regarded,  they  must 
fulfil  the  condition,  that,  when  plotted,  they  shall  form  a  closed 
figure. 


Fig.l. 


Errors  of  Observations. 

5.  Constant  errors  are  those  produced  by  well  understood 
causes,  and  which  may  be  removed  from  the  observations  by 
the   application   of  -computed   corrections.      As   such    may  be 


4  TNTRODUCTION.  1. 

mentioned  :  theoretical  corrections,  like  the  effect  of  tempera- 
ture upon  the  length  of  rods  used  in  measuring  a  base-line  ; 
instrumental  corrections,  like  those  arising  from  a  known  dis- 
crepancy between  the  length  of  the  rods  and  the  standard  of 
measure ;  and  personal  corrections,  like  those  due  to  the  habits 
of  the  observer,  who,  in  making  a  contact  of  the  rods,  might 
err  each  time  by  the  same  constant  quantity.  Strictly  speak- 
ing, then,  constant  errors  are  not  errors ;  since  they  can  always 
be  eliminated  from  the  observations,  when  the  causes  that  pro- 
duce them  are  understood.  The  first  duty  of  an  observer,  after 
taking  his  measurements,  is  to  discuss  them,  and  apply  as  far 
as  possible  the  computed  corrections,  to  remove  the  constant 
errors. 

6.  Mistakes  are  errors  committed  by  inexperienced  and  occa- 
sionally by  the  most  skilled  observers,  arising  from  mental 
confusion.  As  such  may  be  mentioned  :  mistakes  in  reading 
a  compass-needle  by  noting  58°  instead  of  42°;  or  mistakes  in 
measuring  an  angle  by  sighting  at  the  wrong  signal.  Such 
errors  often  admit  of  correction  by  comparison  with  other  sets 
of  observations. 

7.  Accidental  errors  are  those  that  still  remain  after  all  con- 
stant errors  and  all  evddent  mistakes  have  been  carefully  inves- 
tigated, and  eliminated  from  the  numerical  results.  Such,  for 
example,  are  the  errors  in  levelling  arising  from  sudden  expan- 
sions and  contractions  of  the  instrument,  or  from  effects  of 
the  wind,  or  from  the  anomalous  and  changing  refraction  of  the 
atmosphere.  More  than  all,  however,  such  errors  arise  from 
the  imperfections  of  the  touch  and  sight  of  the  observer ;  which 
render  it  impossible  for  him  to  handle  his  instruments  deli- 
cately, estimate  accurately  bisections  of  signals  and  small  divis- 
ions of  graduation,  or  keep  them  continually  in  adjustment. 
These  are  the  errors  that  appear  in  all  numerical  observations, 
however  carefully  the  measurements  be  made,  and  whose  elimi- 


§  8.  £A'A'0/^S   OF  OBSERVATIONS.  5 

nation  is  the  object  of  the  Method  of  Least  Squares.  Al- 
though at  first  sight  it  might  seem  that  such  irregular  errors 
could  not  come  within  the  province  of  mathematical  investiga- 
tion, it  will  be  seen  in  the  sequel  that  they  are  governed  by  a 
wonderful  and  very  precise  law,  namely,  the  law  of  proba- 
bility. 

8.  The  word  "error,"  as  used  in  the  following  pages,  means 
an  accidental  error  produced  by  causes  which  are  numerous,  and 
whose  effects  cannot  be  brought  within  the  scope  of  physical 
investigation.  This  error  is  the  difference  between  the  true 
value  of  the  observed  quantity  and  the  result  of  the  measure- 
ment upon  it.  Thus,  if  Z  be  the  true  value  of  an  angle,  and 
J/,,  M^,  and  M.^  be  the  results  of  measurements  made  upon  it, 
the  differences  Z  —  J/,,  Z  —  M^,  and  Z  —  M^  are  the  errors. 
An  error  is  denoted  by  the  letter  x,  and  subscripts  are  applied 
to  it  for  particular  errors;  thus,  in  the  above  case,  Z — J/,  ^=^x^, 
Z —  M^  =r  x\,  and  Z —  M.^  ==  x\,  or,  in  general,  x  is  the  error  of 
the  observation  M. 

A  residual  is  the  difference  between  the  most  probable  value 
of  the  observed  quantity  and  the  measurement  upon  it.  This 
most  probable  value  is  that  deduced  by  the  application  of  the 
Method  of  Least  Squares  to  the  observations ;  for  instance,  in 
the  simple  case  of  direct  measurements  on  a  single  quantity, 
the  arithmetical  mean  is  the  most  probable  value.  The  residual 
is  denoted  in  general  by  the  letter  v.  Thus,  if  z  be  the  most 
probable  value  of  an  angle  derived  from  the  measurements  M^, 
M^,  and  AT^,  the  residuals  are  a  —  J/,  =  z\,  s  —  J/,  ^=  v-,,  and 
-::  —  J/3  ■=  Vy  Evidently  the  most  probable  value,  .z,  will  ap- 
proach more  nearly  to  the  true  value  Z,  the  greater  the  number 
of  observations,  as  likewise  the  residuals  v  to  the  errors  x. 
With  an  infinite  number  of  precise  observations,  -cr  should  coin- 
cide with  Z,  and  each  v  with  the  corresponding  x.  With  a 
large  number  of  observations,  the  differences  between  the  resid- 


D  INTRODUCTION.  I. 

uals  and  the  errors  will  be  small,  so  that  the  laws  governing  the 
two  will  be  essentially  the  same.  On  this  account  residuals  are 
often  called  residual  errors,  or  sometimes  even  errors. 

Principles  of  Probability. 

9.  The  word  "probability,"  as  used  in  mathematical  reasoning; 
means  a  number  less  than  unity,  which  is  the  ratio  of  the  num- 
ber of  ways  in  which  an  event  may  happen  or  fail,  to  the  total 
number  of  possible  ways  ;  each  of  the  ways  being  supposed 
equally  likely  to  occur.  Thus,  in  throwing  a  coin,  there  are 
two  possible  cases  :  either  head  or  tail  may  turn  up,  and  one  is 
as  likely  to  occur  as  the  other ;  hence  the  probability  of  throw- 
ing a  head  is  expressed  by  the  traction  \,  and  the  probability  of 
throwing  a  tail  also  by  |.  So,  in  throwing  a  die,  there  are  six 
cases  equally  likely  to  occur,  one  of  which  may  be  the  ace  : 
hence  the  probability  of  throwmg  the  ace  in  one  trial  is  \, 
and  the  probability  of  not  throwing  it  is  |. 

In  general,  if  an  event  may  happen  in  a  ways,  and  fail  in  b 
ways,  and   each  of  these  ways  is  equally  likely  to  occur,  the 

probability  of  its  happening  is  ■ -,  and  the  probability  of  its 

failing  is -.     Thus  probability  is  always  expressed  by  an 

a  -\-  b 

abstract  fraction,  and  is  a  numerical  measure  of  the  degree  of 
confidence  which  one  has  in  the  happening  or  failing  of  an 
event.  As  this  measure  may  range  from  o  to  i,  so  mental  con- 
fidence may  range  from  impossibility  to  certainty.  If  the  frac- 
tion is  o,  it  denotes  impossibility ;  if  \,  it  denotes  that  the 
chances  are  equal  for  and  against  the  happening  of  the  event ; 
and  if  i,  the  event  is  certain  to  occur. 

10.  Unity  is  hence  the  mathematical   symbol  for  certainty. 
And,  since  an  event  must  either  happen  or  not  happen,  the  sum 


§   12.  PRINCIPLES   OF  PROBABILITY.  7 

of  the  probabilities  of  happening  and  failing  is  unity.  Thus,  if 
P  be  the  probability  that  an  event  will  happen,  i  —  P  is  the 
probability  of  its  failing.  For  example,  if  the  probability  of 
drawing  a  prize  in  a  lottery  is  2  oV^'  ^^^  probability  of  not  draw- 
ing a  prize  is  \%%%,  a  large  fraction. 

II.  When  an  event  may  happen  in  different  independent 
ways,  the  probability  of  its  happening  is  the  sum  of  the  separate 
probabilities.  For  if  it  may  happen  in  a  ways,  and  also  in  a' 
ways,  and  there  are  c  total  ways,  the  probability  of  its  occur- 
rence (by  Art.  9)  is  ''LJTJL  ;  and  this  is  equal  to  the  sum  of  the 

c 

probabilities  -  and  - ,  of  happening  in  the  separate  independent 
c  c 

ways. 

For  example,  if  there  be  in  a  bag  twenty  red,  sixteen  white, 
and  fourteen  black  balls,  and  one  is  to  be  drawn  out,  the  proba- 
bility that  it  will  be  red  is  ^,  that  it  will  be  white  is  J-^,  and 
that  it  will  be  black  is  '*.  If,  however,  there  be  asked  the 
probability  of  drawing  either  a  red  or  black  ball,  the  answer  is 

zo      I      14    34 

50      '      50  50* 

X2.  A  compound  event  is  one  produced  by  the  concurrence 
of  several  primary  or  simple  events,  each  being  independent  of 
the  other.  For  instance,  throwing  three  aces  with  three  dice 
in  one  trial  is  a  compound  event  produced  by  the  concurrence 
of  three  simple  events.  An  error  of  observation  may  be  re- 
garded as  a  compound  event  produced  by  the  combination  of 
all  the  small  independent  errors  of  the  numerous  accidental 
influences. 

The  probability  of  the  happening  of  a  compound  event  is 
the  product  of  the  probabilities  of  the  several  primary  inde- 
pendent events.  To  show  this,  consider  two  bags,  one  of  which 
contains  seven  black  and  nine  white  balls,  and  the  other  four 


8  INTRODUCTION,  I 

black  and  eleven  white  balls.  The  probability  of  drawing  a 
black  ball  from  the  first  bag  is  ^,  and  that  of  drawing  one  from 
the  second  ^.  What,  now,  is  the  probability  of  the  compound 
event  of  securing  two  black  balls  when  drawing  from  both  bags 
at  once  ?  Since  each  ball  in  the  first  bag  may  form  a  pair  with 
each  one  in  the  second,  there  are  i6  X  15  possible  ways  of 
drawing  two  balls  ;  and,  since  each  of  the  seven  black  balls  may 
be  associated  with  each  of  the  four  black  balls  to  form  a  pair, 
there  are  7X4  cases  favorable  to  drawing  two  black  balls. 
The  required  probability  is  hence  75-^ ;  and  this  is  equal  to 
^  X  ^,  or  the  product  of  the  probabilities  of  the  two  primary 
independent  events. 

To  discuss  the  principle  more  generally,  consider  two  primary 
events,  the  first  of  which  may  happen  in  ^,  ways,  and  fail  in  b^ 
ways,  and  the  second  happen  in  a^,  and  fail  in  b^  ways.  Then 
there  are  Tor  the  first  event  a^  -\-  b,  possible  cases,  and  for  the 
second  a-^  +  b^ ;  and  each  case  out  of  the  a,  -\-  b,  cases  may  be 
associated  with  each  case  out  of  the  a^  +  b^  cases  ;  and  hence 
there  are  for  the  two  events  {a^  -j-  b^)  {a^  +  b^)  total  cases,  each 
of  which  is  equally  likely  to  occur.  In  a^a^  of  these  cases  both 
events  happen  ;  in  b.b^  both  fail ;  in  a^b^  the  first  happens,  and 
the  second  fails  ;  and  in  a^b,  the  first  fails,  and  the  second  hap- 
pens. Hence  (by  Art.  9)  the  probabilities  of  the  compound 
events  are  — 

Probability  that  both  happen  .     .     .     .     . 

Probability  that  both  fail 

Prob.  that  first  happens,  and  second  fails   . 

Prob.  that  first  fails,  and  second  happens   .    ^  ,  ,  ^ 


a,a^ 

(^, 

+  ^.) 

(^. 

a.b:, 

+  b,) 

{a, 

:  +  b,)  {a. 

+  ^.) 

§   14-  PRrNCIPLES   OF  PROPARILITY.  9 

As  each  of  these  probabilities  is  the  product  of  the  proba- 
bilities of  the  primary  events,  the  principle  is  established  for 
the  case  of  two  primary  events.  And  evidently  its  extension  to 
three  or  more  is  easy. 

Thus,  if  there  be  four  events,  and  P ,,  P^,  P^,  and  P^  be  the 
respective  probabilities  of  happening,  the  probability  that  all 
the  events  will  happen  is  P,  P^  P^  P^ ;  and  the  probability  that 
all  will  fail  is  (i  -  P,)  (i  -  A)  (i  -  P,)  (i  -  P,)-  The  prob- 
ability that  the  first  happens  and  the  other  three  fail  is 
P,{\  -  A)  (I  -  P,)  (I  -  P,)\  and  so  on. 

13.  The  most  probable  event  among  several  is  that  which 
has  the  greatest  mathematical  probability.  Thus,  if  two  coins 
be  thrown  at  the  same  time,  there  may  arise  the  three  follow- 
ing compound  cases,  having  the  respective  probabilities  as 
annexed : 

Both  may  be  heads 4 

One  head,  and  the  other  tail 2" 

Both  tails \ 

Here  the  case  of  one  head  and  the  other  tail  has  the  greatest 
probability,  and  is  hence  the  most  probable  of  the  three  com- 
pound events.  The  sum  of  the  three  probabilities,  \,  ^,  and  \, 
is  unity ;  as  should  be  the  case,  since  one  of  these  events  is 
certain  to  occur. 

If  four  measurements  of  the  length  of  a  line  give  the  values 
720.2,  720.3,  720.4,  and  720.5  feet,  the  arithmetical  mean, 
720.35  feet,  is  universally  recognized  as  the  most  probable 
value  of  the  length  of  the  line.  It  will  be  shown  in  the  sequel 
that  the  mathematical  probability  of  this  result  is  greater  than 
of  any  other. 

14.  A  compound  event,  composed  of  any  number  of  simple 
events,  will  now  be  considered.     Let  P  be  the  probability  of 


lO  INTRODUCTION.  I. 

the  happening  of  an  event  in  one  trial,  and  Q  the  probabihty 

of  its  failing,   so  that  P  ^  Q  —  \  :    and    let    there  be  ;/  such 

events.     Then  (by  Art.  12)  the  probability  that  all  will  happen 

is  P" ;   the  probability  that  one  assigned  event   will    fail,  and 

n  —  I   happen,  is  P"~^Q ;  and,  since  this  may  occur  in  ;/  ways, 

the  probability  that  one  will  fail,  and  //  —  i  happen,  is  iiP"~'Q. 

Similarly,  the  probability  of  two  assigned  events  failing,  and 

n  —  2  happening,  is  P'^-'^Q^ ;    and,  since  this  may  be  done  in 

nCn  —  i)  ^     ,  ,    ,  •,•         , 

ways,*  the    probability  that   two    out    of   the  whole 

"J?  (71  — —   T  I 

number  will  fail,  and  ;/ —  2  happen,  is  — ~P"~''Q^.      If, 

2 

then,  {P  +  Q)"  be  expanded  by  the  binomial  formula,  thus, 

(P+  Q)»  =  Pn  +  nP"-'Q  +  ''^^^~   ^^Pn-2Q2  _,_    ,  _ 

1.2 
11  ( n  —  i)  in  —  2)  .  .  .  (n  —  VI  +  i)   ^  _ 

+  -^ — ^ Lpn-mQm  _f_  etc., 

1.2.3  •  '  •  f^l- 

the  first  term  is  the  probability  that  all  will  happen  ;  the  second, 
that  ;/  —  I  will  happen,  and  i  fail  ;  and  the  ni  -\-  i"^  term  is  the 
probability  that  >i  —  ni  will  happen,  and  711  fail.  To  determine, 
then,  the  most  probable  case,  it  is  only  necessary  to  find  the 
term  in  this  series  which  is  greatest. 

The  particular  instance  when  P  ^  (?  =  J  corresponds  to  the 
case  of  throwing  n  coins.     Then  the  series  becomes 


'& 


(i)"  +  ^C^)"  +  ^ {hY  +  -!^ '-^ '  (i)«  +  .  .  ., 

1.2  1-2.3 

in  which  the  middle  term   is  the  greatest  if  ;/  be  even,  and 

*  See  the  theory  of  combinations  in  any  algebra. 


§  '5- 


PRIXCTPLES   OF  PROBABILITY. 


II 


which  has  two  equal  middle  terms  if  n  be  odd.     Thus,  if  n 
the  series  is    ■ 


=  6, 


64  64 


+ 


L5 
64 


64 


64      64      64 


Hence,  if  six  coins  be  thrown,  the  probabilities  of  the  different 
cases  are  the  following  : 

All  heads -gV 

Five  heads  and  one  tail -^ 

Four  heads  and  two  tails |^ 

Three  heads  and  three  tails \% 

Two  heads  and  four  tails    .  " W 

One  head  and  five  tails -ix 

All  tails -^V 

The  sum  of  these  seven  probabilities  is,  of  course,  unity. 

15.  The  following  graphical  illustration  gives  a  clear  view  of 
the  relative  values  of  the  respective  probabilities  of  the  seven 
cases  that  may  arise  in 
throwing  six  coins.  A 
horizontal  straight  line 
is  divided  into  six  equal 
parts,  and  at  the  points 
of  division,  ordinates  are 
erected  proportional  to 
the  probabilities  g'^,  -§^, 
etc.,  and  through  their  extremities  a  curve  is  drawn.  On  the 
same  diagram  is  shown,  by  a  broken  curve,  the  probabilities  of 
the  nine  cases  that  may  arise  in  throwing  eight  coins,  or  the 

terms 

8       .      28      .      56      ,      70 


Fig.  2. 


256  256  256 


+  --+—-+,  etc., 
256         256 


which  are  found  by  expanding  the  binomial  i\  +  .|) 


IA8 


12  INTRODUCTION.  I. 

It  is  one  of  the  weaknesses  of  the  human  mind  that  large 
and  small  numbers  do  not  convey  to  it  accurate  ideas  unless 
aided  by  concrete  analogy  or  representation.  The  above  graphi- 
cal illustration  shows  more  clearly  than  the  numbers  them- 
selves can  do  the  relative  probabilities  in  the  two  cases.  These 
curves,  moreover,  are  very  similar  to  a  curve  hereafter  to  be 
discussed,  which  represents  the  law  of  probability  of  errors 
of  observations. 

Problems. 

i6.  At  the  end  of  each  chapter  will  be  given  a  few  questions 
and  problems.  The  following  will  serve  to  exemplify  the  above 
principles  of  probability  : 

1.  What  is  the  probability  of  throwing  an  ace  with  a  single  die  in 
two  trials?  Ans.  \\. 

2.  A  bag  contains  three  red,  four  white,  and  five  black  balls.  Re- 
quired the  probability  of  drawing  two  red  balls  in  two  drawings,  the  ball 
first  drawn  not  being  replaced  before  the  second  trial  ? 

3.  Each  student  in  a  class  of  twenty  is  likely  to  solve  one  problem 
out  of  every  eight.  What  is  the  probability  that  a  given  problem  will 
be  solved  in  the  class? 

4.  What  is  the  probability  of  throwing  two  aces,  and  no  more,  in  a 
single  throw  with  six  dice?  What  is  the  probability  of  throwing  at  least 
two  aces? 

5.  Let  a  hundred  coins  be  thrown  up  each  second  by  each  of  the 
inhabitants  of  earth.  How  often  will  a  hundred  heads  be  thrown  in  a 
million  years? 

6.  A  purse  contains  nine  dimes  and  a  nickel.  A  second  purse  con- 
tains ten  dimes.  Nine  coins  are  taken  from  the  first  purse  and  put  into 
the  second,  and  then  nine  coins  are  taken  from  the  second  and  put 
into  the  first.     Which  purse  has  the  highest  probable  value  ? 


1 8,  AXIOMS  DERIVED   FROM  EXPERIENCE.  1 3 


CHAPTER   II. 

LAW    OF    PROBABILITY    OF    ERROR. 

17.  The  probability  of  an  assigned  accidental  error  in  a  set 
of  measurements  is  the  ratio  of  the  number  of  errors  of  that 
magnitude  to  the  total  number  of  errors.  It  is  proposed,  in  this 
chapter,  to  investigate  the  relation  between  the  magnitude  of 
an  error  and  its  probability. 

Axioms  derived  from  Experienee. 

18.  An  analogy  often  referred  to  in  the  Method  of  Least 
Squares  is  that  between  bullet-marks  on  a  target  and  errors  of 
observations.  The  marksman  answers  to  an  observer ;  the  posi- 
tion of  a  bullet-mark,  to  an  observation  ;  and  its  distance  from 
the  centre,  to  an  error.  If  the  marksman  be  skilled,  and  all 
constant  errors,  like  the  effect  of  gravitation,  be  eliminated  in 
the  sighting  of  the  rifle,  it  is  recognized  that  the  deviations  of 
the  bullet-marks,  or  errors,  are  quite  regular  and  symmetrical. 
First,  it  is  observed  that  small  errors  are  more  frequent  than 
large  ones  ;  secondly,  that  errors  on  one  side  are  about  as 
frequent  as  on  the  other ;  and,  thirdly,  that  very  large  errors  do 
not  occur.  Further:  it  is  recognized,  that,  the  greater  the  skill 
of  the  marksman,  the  nearer  are  the  marks  to  his  point  of  aim. 

For  instance,  in  the  Report  of  the  Chief  of  Ordnance  for 
1878,  Appendi.x  S',  Plate  VI,  is  a  record  of  one  thousand  shots 
fired  deliberately  (that  is,  with  precision)  from  a  battery-gun,  at 
a  target  two  hundred  yards  distant.     The  target  was  fifty-two 


14 


LAW  OF  PROBABILITY  OF  ERROR. 


II. 


feet  long  by  eleven  feet  high,  and  the  point  of  aim  was  its  cen- 
tral horizontal  line.  All  of  the  shots  struck  the  target ;  there 
being  few,  however,  near  the  upper  and  lower  edges,  and  nearly 
the  same  number  above  the  central  horizontal  line  as  below  it. 
On  the  record,  horizontal  lines  are  drawn,  dividing  the  target 
into  eleven  equal  divisions ;  and  a  count  of  the  number  of  shots 
in  each  of  these  divisions  gives  the  following  results  : 

In  top  division i  shot 

In  second  division 4  shots 

In  third  division 10  shots 

In  fourth  division .89  shots 

In  fifth  division 190  shots 

In  middle  division 212  shots 

In  seventh  division 204  shots 

In  eighth  division.     .     .     .     „ 193  shots 

In  ninth  division 79  shots 

In  tenth  division 16  shots 

In  bottom  division 2  shots 


Total 1,000  shots 

On  Fig.  3  is  shown,  by  means  of  ordinates,  the  distribution  of 

these  shots  ;  A  being  the  top 
division,  B  the  middle,  and  C 
the  bottom  division.  It  will  be 
observed  that  there  is  a  slight 
preponderance  of  shots  below 
the  centre,  and  there  is  reason 
to  believe  that  this  is  due  to 
a  constant  error  of  gravitation 
not  entirely  eliminated  in  the 
sighting  of  the  gun. 

Fig,  3.  19.    The    distribution    of    the 

errors  or  residuals  in   the  case 
of  direct  observations  is  similar  to  that  of  the  deviations  just 


§  21.  7'HE   PROBABILITY  CURVE.  1 5 

discussed.  For  instance,  in  the  United  States  Coast  Survey 
Report  for  1854,  p.  *9i,  are  given  a  hundred  measurements  of 
angles  of  the  primary  triangulation  in  Massachusetts.  The 
residual  errors  (Art.  8)  found  by  subtracting  each  measurement 
from  the  most  probable  values  are  distributed  as  follows  : 

Between  +6".oand  +5".o i  error 

Between  +5.0  and  +4.0 2  errors 

Between  +4-0  and  +3.0 2  errors 

Between  +3.0  and  +2.0 3  errors 

Between  +2.0  and  +1.0 13  errors 

Between  +1.0  and      0.0 26  errors 

Between      0.0  and  —  i.o 26  errors 

Between  —  i.o  and  —2.0 17  errors 

Between  —2.0  and  —3.0 8  errors 

Between  —3.0  and  —4.0 2  errors 

Total 100  errors 


Here  also  it  is  recognized  that  small  errors  are  more  frequent 
than  large  ones,  that  positive  and  negative  errors  are  nearly 
equal  in  number,  and  that  very  large  errors  do  not  occur.  In 
this  case  the  largest  residual  error  was  5". 2;  but,  with  a  less 
precise  method  of  observation,  the  limits  of  error  would  evi- 
dently be  wider. 

20.  The  axioms  derived  from  experience  are,  hence,  the  fol- 
lowing : 

Small  errors  are  more  frequent  than  large  ones. 
Positive  and  negative  errors  are  equally  frequent. 
Very  large  errors  do  not  occur. 

These  axioms  are  the  foundation  of  all  the  subsequent  reasoning. 

The  Probability  Curve. 

21.  In  precise  observations,  then,  the  probability  of  a  small 
error  is  greater  than  that  of  a  large  one,  positive  and  negative 


1 6  LAir  OF  PROBABILITY  OF  ERROR.  II. 

errors  are  equally  probable,  and  the  probability  of  a  very  large 
error  is  zero.  The  words  "very  large"  may  seem  somewhat 
vague  when  used  in  general,  although  in  any  particular  case  the 
meaning  is  clear  ;  thus,  with  a  theodolite  reading  to  seconds,  20" 
would  be  very  large,  and  with  a  transit  reading  to  minutes, 
5'  would  be  very  large.  Really,  in  every  class  of  measure- 
ments there  is  a  limit,  /,  such  that  all  the  positive  errors  are 
included  between  o  and  -j-  /,  and  all  the  negative  ones  between 
o  and  —  /. 

22.  Hence  the  probability  of  an  error  is  a  function  of  that 
error ;  so  that,  calling  x  any  error  and  y  its  probability,  the  law 
of  probability  of  error  is  represented  by  an  equation 

and  will  be  determined,  if  the  form  of  f{.x)  can  be  found.  If, 
then,  _>'  be  taken  as  an  ordinate,  and  x  as  an  abscissa,  this  may 
be  regarded  as  the  equation  of  a  curve  which  must  be  of  a  form 
to  agree  with  the  three  fundamental  axioms  ;  namely,  its  maxi- 
mum ordinate  OA  must  correspond  to  the  error  zero  ;  it  must 
be  symmetrical  with  respect  to  the  axis  of  Y,  since  positive  and 
negative  errors  of  equal  magnitude  are  equally  probable  ;  as  x 
increases  numerically,  the  value  of  y  must  decrease,  and,  when 
X  becomes  very  large,  j  must  be  zero.  Fig.  4  represents  such 
a  curve,  OP  and  OM  being  errors,  and  PB  and  MC  their  re- 
spective probabilities.  Further  :  since  different  measurements 
have  different  degrees  of  accuracy,  each  class  of  observations 
will  have  a  distinct  curve  of  its  own. 

The  curve  represented  in  Fig.  4  is  called  the  probability  curve. 
In  order  to  determine  its  equation,  it  is  necessary  to  consider 
y  as  a  continuous  function  of  x.  This  is  evidently  perfectly 
allowable  ;  since,  as  the  precision  of  observations  is  increased, 
the  successive  values  of  x  are  separated  by  smaller  and  smaller 
intervals.     The  requirement  of  the  third  axiom,  that  y  must  be 


§23. 


F/RST  DEDUCTION  OF  THE   LA  IV  OF  ERROR. 


17 


zero  for  all  values  of  x  greater  than  the  limit  ±  /,  is  apparently 
an  embarrassing  one,  as  it  is  impossible  to  determine  a  continu- 
ous function  of  x  which  shall  become  zero  for  x  =z  ±  I  and  also 
be  zero  for  all  values  of  x  from  ±  /  to  ±  00 .  But,  since  this 
limit  /  can  never  be  accurately  assigned,  it  will  be  best  to  extend 
the  limits  to  ±  00 ,  and  determine  the  curve  in  such  a  way  thrt 


the  value  of  j,  although  not  zero  for  large  values  of  ,r,  will  be  so 
very  small  as  to  be  practically  inappreciable.  The  equation  of 
the  probability  curve  will  be  the  mathematical  expression  of  the 
law  of  probability  of  errors  of  observation.  Two  deductions  of 
this  law  will  be  given  ;  the  first  that  of  Hagen,  and  the  second 
that  of  Gauss. 


First  Deduct  ion  of  the  Law  of  Error. 

23.  Hagen's  demonstration  rests  on  the  following  hypothesis 
or  axiom,  derivcJ  from  experience  : 

An  error  is  the  algebraic  sum  of  an  indefinitely  great  number 
of  small  elementary  errors  which  are  all  equal,  and  each  of  which 
is  equally  likely  to  be  positive  or  negative. 

To  illustrate  :  suppose  that,  by  several  observations  with  a 
levelling  instrument  and  rod,  the  difference  in  elevation  between 


1 8  LAW  OF  PROBABILITY  OF  ERROR.  II. 

two  points  has  been  determined.  This  value  is  greater  or  less 
than  the  true  difference  of  level  by  a  small  error,  x.  This  error 
X  is  the  result  of  numerous  causes  acting  at  every  observation  : 
the  instrument  is  not  perfectly  level,  the  wind  shakes  it,  the 
sun's  heat  expands  one  side  of  it,  the  level-bubbles  are  not  accu- 
rately made,  the  glass  gives  an  indistinct  definition,  the  tripoct 
is  not  firm,  the  eye  of  the  observer  is  not  in  perfect  order,  there 
is  irregular  refraction  of  the  atmosphere,  the  man  at  the  rod 
does  not  hold  it  vertical,  the  turning-points  are  not  always  good 
ones,  the  graduation  of  the  rod  is  poor,  the  target  is  not  prop-- 
erly  clamped,  the  rod-man  errs  in  taking  the  reading,  and  many 
others.  Again  :  each  of  these  causes  may  be  subdivided  into 
others  ;  for  instance,  the  error  in  reading  the  rod  may  be  due, 
perhaps,  to  the  accumulated  result  of  hundreds  of  little  causes. 
The  total  error,  x,  may  hence  be  fairly  regarded  as  resulting 
from  the  combination  of  an  indefinitely  great  number  of  small 
elementary  errors  ;  and  no  reason  can  be  assigned  why  one  of 
these  should  be  more  likely  to  be  positive  than  negative,  or 
negative  than  positive. 

24.  Now,  it  is  evident  that  it  is  more  probable  that  the 
number  of  positive  elementary  errors  should  be  approximately 
equal  to  the  number  of  negative  ones  than  that  either  should  be 
markedly  in  excess,  and  that  the  probability  of  the  elementary 
errors  being  either  all  positive  or  all  negative  is  exceedingly 
small.  In  the  first  case  the  actual  error  is  small,  and  in  the 
second  large  ;  and  so  the  probabilities  of  small  errors  are  the 
greatest,  and  the  probability  of  a  very  large  error  is  practically 
zero.  These  correspond  to  the  properties  which  the  proba- 
bility curve  must  possess. 

Let  A.*-  represent  the  magnitude  of  an  elementary  error,  and 
;;/  the  number  of  those  errors.  The  probability  that  any  \x 
will  be  positive  is  \,  and  that  it  will  be  negative  is  also  \.  The 
probability  that  all  of  the  ui  elementary  errors  will  be  positive 


§25. 


FIRST  DEDUCTION  OF   THE   LAW  OF  ERROR. 


19 


is  hence  (1)'"  ;  the  probability  that  in  —  i  will  be  positive  and 
I  negative  is  w(^-)'"~'(|)' ;  and  the  probabilities  of  all  the  re- 
speotive  cases  will  be  given  by  the  corresponding  terms  of  the 
binomial  formula  (Art.  14).  When  all  of  the  ;;/  elementary 
errors  are  positive,  the  resulting  error  of  observation  is  +  w.A.r; 
when  ;//  —  i  are  positive  and  i  negative,  the  resulting  error  is 
-\-  {ui  —  \)\x  —  A.r,  or  -|-  (;;/  —  2)\x.  If  in  —  n  elementary 
errors  are  positive  and  the  remaining  n  are  negative,  the  result- 
ing error  is  -f-  {in  —  ii)\x  —  n.^x,  or  +  {in  —  2n)ii,x,  and  the 
probability  of  this  particular  combination  is  given  by  the 
;^ -f"  i^*  term  of  the  expansion  of  the  binomial  {\-\-\)'".  It  is 
easy  then  to  write  the  following  table  : 


Elementary  Errors  \.x. 


If  m  are  +  and  o  are— 
If  in  —  \  are  +  and  i  is  — 
If  ;;/  —  2  are  +  and  2  are  — 
If  wz  — 3  are  +  and  3  are  — 


Resulting 
Error  x. 


»t\x 

(w  —  2)\X 

(;«  — 4)A.r 
{m  —  6)\x 


Its  Probability  y. 


m\ 


0'" 

m[m—\)  /,\  „i 

1.2         \2/ 
»i{m  —  \){m  —  2)  /i\  1,1 


1.2.3 


C) 


If  vt  —  n  are  +  and  ti  are  — 

If  m  —  n  —  i  are  f  and  «  +  i  are  - 


(m  —  2n)Ax 
(>n  —  2n  —  2)^x 


m(m  —  i)(w— 2)  .  .  .  (m  —  n+  i)  /I 


0 


w(w  —  I )(/«  —  2)  .  .  .  (w  — ;? )  /I 


1.2.3 


(B 


25.  In  the  curve  y  =/{x)  let  OM  be  any  error  x,  and  AfC 
its  probability  J//  also  let  OP  be  an  error  ^  less  in  magnitude, 
and  PB  its  corresponding  probability  y.     Then,  from  the  figure. 


,.    .    BD        ,.    .   y-y         dy 

limit  =  limit —,  =   — 

CD  X  -  X         dx 


20 


LAW   OF   PROBABILITY    OF    ERROR. 


II. 


is  the  differential  equation  of  the  curve.      To  deduce,  then,  the 


law  of  probability  of  error,  it   is  only  necessary  to  find 


y—y 


ill  terms  of  y  and  x,  pass  to  the  limit,  place  it  equal  to  y-,  and 

(IOC 

perform  the  integration. 

If  x'  be  taken  as  the  error  next  less  in   magnitude  to  x,  the 

y  —  y' 
difference  x—x    equals  2A.v,  and   the   value   of ~   is  the 

dy 
limit  -r  if  the  curve  is  to  be  continuous. 
ax 


26.   For  the  two  consecutive  errors  x  and  x'  take  (from  Art. 
24)  the  two  general  values 

a;  =  (m  —  2n)A.Y,   and   x'  =  {m  —  in  —  2)A.t. 

The  ratio  of  the  probabilities  of  these  errors  is 

y'  _  m  —  n 


y 


«+  i' 


which,  after  inserting  for  n  its  value  in  terms  of  x,  m,  and  Ax, 
may  be  put  into  the  form 

2{i\X  —  x)  —   2X  \ 


y-  y  =  y 


{m  +  2)Ajc  —  X 


-^y 


»zA:K 


§  26.  FIRST  DEDUCT/ON-  OF   THE   LAW  OF  ERROR.  21 

Here  A-v  in  the  numerator  vanishes  in  comparison  with  x.  In 
the  denominator,  2  vanishes  compared  with  in,  and  ;;zAa'  is  the 
maximum  positive  error,  which  is  so  large  that  x  vanishes  in 
comparison  with  it.     The  differential  equation,  then,  is 

(^y  _  y  —  y'  _  yx 

or 

-^  =  —2/iyx, 

dx 

in  which  2]f  has  been  written  to  represent  the  quantity  ~~, 

The  integration  of  this  equation  gives 

log^  =  —h^x^  +  k\ 

in  which  k'  is  the  constant  of  integration,  and  the  logarithm 
is  in  the  Napierian  system.  By  passing  from  logarithms  to 
numbers 

in  which  e  is  the  base  of  the  Napierian  system.  Since  6^  is  a 
constant,  this  may  be  written 

(i)  y=ke-^'^\ 

and  this  is  the  equation  of  the  probability  curve,  or  the  equa- 
tion expressing  the  law  of  probability  of  errors  of  observation. 

This  equation  satisfies  the  conditions  imposed  in  Art,  22, 
for  J  is  a  maximum  when  ,r  is  o;  it  is  symmetrical  with  respect 
to  the  axis  of  Y,  since  equal  positive  and  negative  values  of  x 
give  equal  values  of  y,  and  when  x  becomes  very  large,  y  is 
very  small.  The  constants  k  and  h  will  be  particularly  consid- 
ered hereafter. 


22  LAW  OF  PROBABILITY  OF  ERROR.  II. 

Second  Deduction  of  the  Lcnu  of  Error. 

27.  Gauss's  demonstration  is  based  on  the  following  hypoth- 
esis or  axiom,  established  by  experience  : 

The  most  probable  value  of  a  quantity  which  is  observed 
directly  several  times,  with  equal  care,  is  the  arithmetical  mean 
of  the  measurements. 

The  average  or  arithmetical  mean  has  always  been  accepted 
and  used  as  the  best  rule  for  combining  direct  observations  of 
equal  precision  upon  one  and  the  same  quantity.  This  universal 
acceptance  may  be  regarded  as  sufficient  to  justify  the  axiom 
that  it  gives  the  most  probable  value,  the  words  "most  prob- 
able" being  used  in  the  sense  of  Art.  13;  for  after  all,  as 
Laplace  has  said,  the  theory  of  probability  is  nothing  but  com- 
mon sense  reduced  to  calculation.  If  the  measurements  be  but 
two  in  number,  the  arithmetical  mean  is  undoubtedly  the  most 
probable  value  ;  and,  for  a  greater  number,  mankind,  from  the 
remotest  antiquity,  has  been  accustomed  to  regard  it  as  such. 

It  is  a  characteristic  of  the  arithmetical  mean  that  it  renders 
the  algebraic  sum  of  the  residual  errors  zero.  To  show  this,  let 
J/„  M^  .  .  .  M„,  be  n  measurements  of  a  quantity ;  then  the 
arithmetical  mean  of  these  is, 

M,  +  M,  +  yl/3  +  .  .  .  +  M„ 

z  = • 

n 

This  equation  may  be  written 

nz  =  M,  -}-  J/,  +  J/3  +  .  .  .  +  M„, 
which  by  transposition  becomes 

(2  -  M,)  +  (s  -  M,)  +  (z  -  J/3)  -f  .  .  .  +  (2  -  J/«)  =  o : 
that  is  to  say,  the  arithmetical  mean  requires  that  the  algebraic 


§28.        SECOND   DEDUCTION  OF   THE  LAW  OF  ERROR.  23 

sum  of  the  residual  errors  shall  be  zero.  To  take  a  numerical 
illustration,  let  730.4,  730.5,  and  730.9  be  three  measure- 
ments  of  the  length  of  a  line.  The  arithmetical  mean  is  730.6, 
giving  the  residuals  +0.2,  +0.1,  and  —0.3,  whose  algebraic 
sum  is  o 

28.  Consider  the  general  case  of  indirect  observations,  in 
which  it  is  required  to  find  the  most  probable  values  of  quanti- 
ties Ly  measurements  on  functions  of  those  quantities.  For 
simplicity,  only  two  quantities,  .c,  and  z^,  will  be  considered; 
although  the  reasoning  is  general,  and  applies  to  any  number. 
Let ;/  observations  be  made  on  functions  of  xr,  and  z^,,  from  which 
it  is  required  to  find  the  most  probable  values  of  z^  and  z^.  The 
differences  between  the  observations  and  the  corresponding  true 
values  of  the  functions  are  errors  ,r„  x^,  .  .  .  x„,  each  of  which  is 
also  a  function  of  z,  and  z^.    The  probabilities  of  these  errors  are 

And  by  Art.  12  the  probability  of  committing  the  given  system 
of  errors  is 

^  =  yo'^yi  •  •  • ;«  =  /(-^O  A^2)  ■  •  ./(•^«)- 

Applying  logarithms  to  this  expression,  it  becomes 

log  F  =  log/(.r,)  +  log/(x,)  4-  .  .  .  +  log/(^«)- 

Now,  the  most  probable  values  of  the  unknown  quantities  ^r, 
and  z^  are  those  which  render  P  a  maximum  (Art.  13),  and 
hence  the  derivative  of  P  with  respect  to  each  of  these  variables 
must  be  equal  to  zero.  Indicating  the  differentiation,  the  follow- 
ing equations  result : 

Pdz,        /{x,)dz,        f{x,)dz,        "'       /{x„)dz, 

dP    ^    df{x,)     ^    df{x;)     ^     _  ^    ^(^(^«)    ^  o. 
PdZ:,        Ax^)dz^        /{x;)dz^        '  '  '       A^'^dz^ 


24  LA  W  OF  PROBABILITY  OF  ERROR.  II. 

Since  in  general  df{x)  =  (fi(x)/(x)dx,  these  may  be  written 

<l>(x,)'^  +  <t>(x,)'^  +  ...  +  H^-u)"^  =  o, 
azt  dZi  dzi 

<f>(x.)'^'  +  <f^(xy^   +   .  .  .   +   <^(^«)^  =  o, 
az2  az2  dzj, 

and,  being  as  many  in  number  as  there  are  unknown  quantities, 
they  will  determine  the  values  of  those  unknown  quantities  as 
soon  as  the  form  of  the  function  ^  is  known. 

Since  these  equations  are  general,  and  applicable  to  any  num- 
ber of  unknown  quantities,  the  form  of  the  function  0  may  be 
determined  from  any  special  but  known  case.  Such  is  that  in 
which  there  is  but  one  unknown  quantity,  and  the  observations 
are  taken  directly  upon  that  quantity.  Thus,  if  there  be  only 
the  quantity  c,  and  the  measurements  give  for  it  the  values 
i^/„  J/j  .  .  .  yT/„,  the  errors  are, 

3c,  =  2  —  /J/„    X2  =■  z  —  M2  .  .  .  Xn  =■  z  —  M„^ 

from  which 

dx^  _  dx2  dXft 

dz  dz         '  '  '         dz 

and  the  first  equation  above  becomes 

<i>{x,)  +  <t>{x,)  +  ct>{x^)  4-  .  .  .  +  </)(.r„)  =  o. 

In  this  case,  also,  the  arithmetical  mean  is  the  most  probable 
value,  and  the  algebraic  sum  of  the  residuals  will  be  zero,  or,  if 
V  denote  any  residual  in  general, 

z'i  +  z>2  +  Vi  +  .  .  .  4-  z'„  =  o. 

Now,  if  the  number  of  observations,  ;/,  is  very  large,  the  resid- 
uals V  will  coincide  with  the  errors  x  (Art.  8),  and 

Xi  ~j~  X2  ~r  Xt^  -)-•.•  -f-  Xii  =  o« 


§29  DISCUSS/O.V  OF   THE    CURVE  y  =  ke~h^ x^ .  25 

This  equation  can  only  agree  with  that  above  when  ^  signifies 
multiplication  by  a  constant,  or  when 

^{Xi^   +  <A('V,)   4-  .  .  .  -f  '^{x„)  =  cx^  +  ^^2  4-  •  •  .  4-  CXn. 
Replacing  in  this  the  values  of  <^(.r,),  ^{x^,  etc.,  it  becomes 
■       ^/(x.)     ^     df{x.^    ^  ^^^_  ^  ^^_  ^  ^^^  ^  ^^^^ . 


f{x,)dx,        /{x^)dx2 

and,  since  this  is  true  whatever  be  the  number  of  observations, 
the  corresponding  terms  in  the  two  members  are  equal.  Hence, 
if  X  be  any  error,  and  j'  =/(,r), 

df{x)  dv 

^         =    -^—  =  ex. 
/{x)dx        ydx 

Multiplying  both  members  by  dx,  and  integrating, 

\ogy  =  —  +  k', 
2 

Passing  from  logarithms  to  numbers. 

Here  the  constant  c  must  be  essentially  negative,  sinpe  the 
probability  J  should  decrease  as  x  increases  numerically  ;  repla- 
cing it,  then,  by  —2/1-,  and  also  putting  c'''  =  /-,  there  results 

(i)  y='ke-^''^\ 

which  is  the  equation  of  the  probability  curve,  or  the  equation 
expressing  the  law  of  probability  of  errors  of  observation. 

Discussion  of  the  Curve  y  =  ke  ~^^^  ^^. 

29.  Since  positive  and  negative  values  of  x  numerically  equal 
give  equal  values  of  y,  the  curve  is  symmetrical  with  respect  to 
the  axis  of   Y.     The  maximum  value  of  y  is  for  x  =.o,  when 


26 


LAW  OF  PROBABILITY  OF  ERROR. 


II. 


y  ^=  k ;  k  is,  hence,  the  probability  of  the  error  o.  As  ,i'  in- 
creases numerically,  y  decreases  ;  and  when  x  =i  zo,  y  becomes  o. 
The  value  of  the  first  derivative  is 


dx 


—  —2kh'^e-''^^^x, 


which  becomes  zero  when  x  ==  o  and  when  x  =  ±  oo,  indicating 
that  the  curve  is  horizontal  over  the  origin,  and  that  the  axis  of 
X  is  an  asymptote.     The  value  of  the  second  derivative  is 

dy 


dx' 


=  —2kh^e-''^-'^-  (—  2h^x^  +  1)3 


which   becomes  o  when    —2h^x^-\-  i  =:  o,  indicating    that   the 
curve  has  an  inflection-point  when  x  =  ± 


h^z 


To  show  further  the  form  of  the  curve,  the  following  values 
have  been  computed,  taking  k  and  Ji  each  as  unity  : 


I 

X 

y 

X 

y 

.0 

1 .0000 

±1.8 

0.0392 

±0.2 

0.9608 

±2.0 

0.0183 

±0.4 

0.8521 

±2.2 

0.0079 

±0.6 

0.6977 

±2.4 

0.0032 

±0.8 

0-5273 

±2.6 

0.0012 

±1.0 

0.3679 

±2.8 

0.0004 

±1.2 

0.2370 

±3.0 

O.OOOI 

±1.4 

0.1409 

±1.6 

0.0773 

±00 

0.0000 

§31-  THE   PROBABILITY  INTEGRAL.  2^ 

The  curve  in  Fig.  4  is  constructed  from  these  values,  the  ver- 
tical scale  being  double  the  horizontal.  C  is  the  inflection- 
point,  whose  abscissa  Oill  is  0.707. 


30.  The  constant  /i  is  a  quantity  of  the  same  kind  as  \,  since 

the  exponent  h~x^  must  be  an  abstract  number.  Methods  will 
be  hereafter  explained  by  which  its  value  may  be  determined 
for  given  observations.  The  probability  of  an  assigned  error  x/ 
decreases  as  h  increases  ;  and  hence,  the  more  precise  the  ob- 
servations, the  greater  is  h.  For  this  reason  h  may  be  called 
"the  measure  of  precision." 

The  constant  k  is  an  abstract  number ;  and,  since  it  is  the 
probability  of  the  error  o,  it  is  larger  for  good  observations  than 
for  poor  ones.  The  more  precise  the  measurements,  the  larger 
is  k. 

TJie  P7-obability  Litegral. 

31.  To  determine  the  value  of  the  constant  k,  and  also  to 
investigate  the  probability  of  an  error  falling  between  assigned 
limits,  the  following  reasoning  may  be  employed : 

Let  y,  .r„  .\\  ...  a-  be  a  series  of  errors,  x'  being  the  smallest, 
.r,  the  next  following,  and  x  the  last ;  the  differences  between 
the  successive  values  being  equal,  and  x  being  any  error. 
Then,  by  Art.  11,  the  probability  of  committing  one  of  these 
errors,  that  is,  the  probability  of  committing  an  error  lying 
between  x'  and  x,  is  the  sum  of  the  separate  probabilities 
ke'^'^'^y  ^,^, -A-^i^^  ^.|-(,  .  (^j.^  jf  p  tienote  this  sum, 

P=  k{e-h''-^'^  +  e-h^-^^"  ^- e-''^'-^^  +  ...  4-^"^''-^'), 

which  may  be  written 

P  =  kt"^' e- ^''\ 

the   notation   2J.  denoting   summation  from  ^  \.q  x  inclusive. 


28  LAW  OF  PROBABILITY  OF  ERROR.  II. 

To  replace  the  sign  of  summation  by  that  of  integration,  dx 
must  be  the  interval  between  the  successive  values  of  the 
errors,  and  then  the  probability  that  an  error  will  lie  between 
any  two  limits  x'  and  x  is 


dxJjc' 


Now,  it  is  certain  that  the  error  will  lie  between  —  oo  and  -|-  oo , 
and,  as  unity  is  the  symbol  for  certainty, 

I  =  —  /        e-f^'^'dx. 

dx  J  — CO 

The  value  of  the  definite  integral  in  this  expression  is  —.* 
Hence 


I  = 


Jidx* 


*  The  following  method  of  determining  this  integral  is  nearly  that  presented  by 
Sturm  in  his  Cours  d'Analyse,  Paris,  1857,  vol.  ii.  p.  16. 

The  integral  fe~^''-^'Jx  e.xpresses  the  area  between  the  probability  curve  and 
the  axis  of  X,  and,  since  the  curve  is  symmetrical  to  the  a.xis  of  V,  that  integral 
between  the  limits  —  <»  and  +  00  will  be  equal  to  double  the  integral  between  the 
limits  o  and  +  00 .     Placing  also  /ix  —  /, 

/+  00  'J      /•oo 

and  the  integral  in  the  second  member  is  to  be  deterinined. 

Take  three  co-ordinate  rectangular  axes  OT,  Oil,  and  OF,  and  change  /  into  «, 
then 

JfQO         _ 
e     *'Jt  —  area  between  curve  F/T'and  axes, 
o 

/•oo 

A  =   I     e  ~  "^du  =  area  between  curve  VuUznd  axes. 

Jo 
/•oo    /•oo 

and  ^^=1  X  ^  ""''-"  ^'«'«- 


^  ^2.  THE   PROBABILITY  INTEGRAL.  29 

from  which  the  value  of  k  is 

hdx 
~  ^^' 
The  equation  of  the  probabiUty  curve  now  becomes 

(2)  y  =  /idxTr~ie~^''^% 

and  the  probabiHty  that  an  error  Hes  between  any  two  given 
Hmits  x'  and  x  becomes 

Equations    (i),  (2),  and  (3)  are   the  fundamental  ones  in  the 
theory  of  accidental  errors  of  observation. 

32.  The  probability  that  an  error  hes  between  the  limits  —  x 
and  +,r  is  double  the  probability  that  it  lies  between  the  limits 
o  and  -\-x,  on  account  of  the  symmetry  of  the  curve.     Hence 

2/1 

7' 


<3)  ^=7^//''"'*^- 


(4)  j>^iijy..,. 


Now  V  =  e~  <^  is  the  equation  of  the  curve  FtT,  and  v  =  e-u^  \s  the  equation 
of  t^uL^,  and,  if  either  of  these  curves  revolves  about  the  axis  of  F,  it  generates  a 
surface  whose  equation  is  v  —  e~i^-"''.  Hence  the  double  integral  A^  \s  one- 
fourth  of  the  volume  included  between  that  surface  and  the  horizontal  plane.  If  a 
series  of  cylinders  concentric  with  the  axis  r  form  the  volume,  the  area  of  the  rmg 
included  between  two  whose  radii  are  r  and  r  +  dr  is  z-nrdr,  and  the  corresponding 
height  \sv  —  e-  t^-  ■u'^  —  e-i"^.     Hence  one-fourth  of  the  volume  is 

A^  ^  -  I     e  ~  ^^  z-Krdr, 

which,  s'mcej e~''^2rdr  z=  e  -r^,  is  equal  to  -.     Therefore 

•'o  2 

and  hence,  finally, 

/  +  CO  2    /•»  i/jj- 


30  LAIV  OF  PROBABILITY  OF  ERROR.  II. 

expresses  the  probability  that  an  error  is  numerically  less  than 
X.     This  may  be  written  in  the  form 

(4)  P^  —  f'^-'^-'^'dihx), 

and  is  called  the  probability  integral. 

As  the  number  of  errors  of  the  magnitude  x  is  proportional 
to  the  probability  y,  and  as  P  in  equation  (4)  is  merely  the 
summation  of  the  probabilities  of  all  errors  between  —  x  and 
+  X,  the  number  of  errors  between  these  limits  is  also  pro- 
portional to  P.  Now,  P  is  the  area  of  the  probability  curve 
between  the  limits  —  x  and  +  x,  the  whole  area  being  unity. 
Hence  the  number  of  errors  between  two  assigned  limits  ought 
to  bear  the  same  ratio  to  the  whole  number  of  errors  as  the 
value  of  P  between  these  limits  does  to  unity. 

By  the  usual  methods  of  the  integral  calculus  the  value  of 
the  probability  integral  corresponding  to  successive  numerical 
values  of  hx  may  be  computed.*  A  table  of  these  values  is 
given  at  the  end  of  this  volume  (Table  I.). 


2    c^ 

*  First  put  hx  =  /,  then  -pj   e~t^dt  is  the  integral  to  be  evaluated.     By  devel- 
oping e-i^  into  a  series  by  Maclaurin's  formula,  the  following  results: 

2  /         fi  I       /s  I        /'  ,  \ 

P  —  -7=.(  / +   —  •  -  —   •  — h  etc.  I, 

VA         31-2     5         I---3     7  /' 

,vhich  is  convenient  for  small  values  of  t.     For  large  values  integrate  by  parts,  thus 

1  I  1  r-?  ~ '" 


-dt 


.-i2 


§33-  COMPARISON  OF   THEORY  AND   EXPERIENCE.  3 1 

To    illustrate    the    use    of   this   table,  consider  the    case   of 

//.r  =  1.24,  for  which  /^  =:  0.9205.      Here  0.9205   is  the   proba- 

I  24 
bility  that  an  error  will  be  numerically  less  than   ~^  \  or,  in 

other  words,  if  there  be  10,000  observations,  it  is  to  be  expect- 

1.24 
ed  that  in  9,205  of  them  the  errors  would  lie  between '~ 

and  A — '-^,  and  in  the  remaining  795  outside  of  these  limits. 
h 


Couiparison  of  Theory  and  Experience. 

33.  By  means  of  Tabic  I  the  theory  employed  in  the  deduc 
tions  of  equations  (i),  (2),  (3),  and  (4)  may  be  tested.  To  use 
the  table  it  is  necessary  to  know  the  value  of  the  constant  //. 
Granting  for  the  present  that  it  may  be  determined,  the  fol- 
lowing examples  will  exemplify  the  accordance  of  theory  and 
experience. 

For  the  one  hundred  residual  errors  discussed  in  Art.  19,  the 

value  of  //  may  be  determined  to  be  -j-. -. 

2  .236 


And  since  ]     e  —t^dt  =  — ,  as  shown  in  the  preceding  footnote, 

•/o  2 


^"      '^.-/v/, 


from  which /'=  I  -^^[i-^  +  ^--P[3  +  etc.J 

From  these  two  series  the  values  of  P  can  be  found  to  any  required  degree  of 
accuracy  for  all  values  of  /  or  Ax. 


32 


LAW  OF  PROBABILITY  OF  ERROR. 


II. 


for  X  =  i".o 

with  /ix  =  0.447 

for  X  —     2.0 

with  /ix  =  0.894 

for  X  =    3.0 

with  /lv  =  1. 34 1 

for  X  =    4.0 

with  /ix  =   1.788 

for  X  =     5 .0 

with  /ix  =  2.235 

for  .r  =      00 

with  /ix  =  00 

Then  from  the  table  the  following  values  of  P  are  taken : 

the  area  P  =  0.473 
the  area  P  =  0.794 
the  area  P  =  0.942 
the  area  P  =  0.989 
the  area  P  —  0.998 
the  area  P  =  i  .000 

Now,  these  probabilities  or  areas  P  are  proportional  to  the 
number  of  errors  less  than  the  corresponding  values  of  x. 
Hence  multiplying  them  by  100,  the  total  number  of  errors, 
and  subtracting  each  from  that  following,  the  number  of  theo- 
retical errors  between  the  successive  values  of  x  is  found. 
The  following  is  a  comparison  of  the  number  of  actual  and 
theoretical  errors  : 


Limits 

Actual  Errors. 

Theoretical 
Errors. 

Differences. 

o".o  and  i".o 

52 

47 

+  5 

i.o  and    2.0 

30 

32 

—  2 

2.0  and    3.0 

II 

15 

-4 

3.0  and    4.0 

4 

5 

—  I 

4.0  and    5.0 

2 

I 

+  1 

5.0  and    6.0 

I 

0 

+  1 

6.0  and    CO 

0 

0 

0 

The  agreement  between  theory  and  experience,  though  not 
exact,  is  very  satisfactory  when  the  small  number  of  observa- 
tions is  considered. 

34.  Numerous  comparisons  like  the  above  have  been  made 
by  different  authors,  and  substantial  agreement  has  always 
been  found  between  the  actual  distribution  of  errors  and  th^ 


§  35. 


JiEMARA'S   O.V   THE   FUNDAMENTAL    FORMULAS. 


33 


theoretical  distribution  required  by  equations  (2)  and  (4).  The 
following  is  a  comparison  by  Besscl  of  the  errors  of  three  hun- 
dred observations  of  the  right  ascensions  of  stars  : 


Limits. 

Actual  Errors. 

Theoretical 
Errors. 

Differences. 

0^.0  and  o^.i 

114 

107 

+  7 

0.1  and   0.2 

84 

87 

-3 

0.2  and   0.3 

53 

57 

-4 

0.3  and   0.4 

24 

30 

-6 

0.4  and   0.5 

14 

13 

+  1 

0.5  and   0.6 

6 

5 

+  1 

0.6  and   0.7 

3 

I 

-1-2 

0.7  and   0.8 

I 

0 

+  1 

0.8  and   0.9 

I 

0 

+  1 

0.9  and    CO 

0 

0 

0 

The  differences  are  here  relatively  smaller  than  in  the  previous 
case.  And  in  general  it  is  observed  that  the  agreement  be- 
tween theory  and  experience  is  closer,  the  greater  the  number 
of  errors  or  residuals  considered  in  the  comparison. 

Whatever  may  be  thought  of  the  theoretical  deductions  of 
the  law  of  probability  of  error,  there  can  be  no  doubt  but  that 
its  practical  demonstration  by  experience  is  entirely  satisfac- 
tory. 

Remarks  on  the  licndamental  Fonnulas. 

35.    The  two  equations  of  the  probability  curve, 

(i)  y  =  ke-'^'-\ 

(2)  y  =  h.dx.-K~\e~  ^'''''^, 

are   identical,  and   the   former   has    already   been     discussed  at 


34  LA  IV  OF  PROBABILITY  OF  ERROR.  11. 

length.  In  the  latter,  dx  for  any  special  case  is  the  interval 
between  successive  values  of  x.  For  instance,  if  observations 
of  an  angle,  be  carried  to  tenths  of  seconds,  dx  is  o".  i  ;  if  to 
hundredths  of  seconds,  dx  is  o'^oi  ;  and  if  a  continuous  curve 
is  considered,  dx  is  the  differential  of  x.  As  y  is  an  abstract 
number,  Ji.dx  must  likewise  be  abstract,  and  hence  h  must  be  a 

quantity  of  the  same  kind  as  — .     The  probability  of  the  error 

dx 

o  is  —^  ;  thus  in  measuring  angles  to  hundredths  of  seconds, 

1            1    1  •!•        1                          •       //        •     o".o\h       .       ,  .     . 
the  probabnity  that   an   error  is  o  .09  is — .     As  this  in- 

V/tt 

creases  with  //,  the  value  of  h  may  be  regarded  as  a  measure  of 
the  precision  of  the  observations.  Methods  of  determining  k 
are  given  in  Chap.  IV. 

36.    The  two  probability  integrals, 

h 


(3)  ^=vcX"~'^"'^'^' 

(4)  F^-^-j'%-^'^-'-dhx, 


are  identical,  exxept  in  their  limits.  The  first  gives  the  proba- 
bility  that  an  error  will  lie  between  any  two  limits  ,f'and  x ;  and 
the  second,  the  probability  that  it  lies  between  the  limits  —  x 
and  -\-  X,  or  that  it  is  numerically  less  than  x.  The  second  is 
then  a  particular  case  of  the  first.  Table  I  refers  only  to  (4)  ; 
and  from  it  by  simple  addition  or  subtraction  the  probability 
can  be  found  for  any  two  assigned  limits.  For  example,  the 
probability  that  an  error  lies  between  —  2^.0  and  +  ^".o  is  the 
sum  of  the  probabilities  for  the  limits  o".o  to  2'''.o  and  o".o  to 
4".o ;  and  the  probability  that  an  error  is  between  +  ^".o  and 
+  4".o  is  Ihe  difference  of  the  probabilitiea  of  those  limits. 

The  integral  P  is  simply  the  summation  of  the  val  :es  of  j 


^  S7-  PROBLEMS  AND    QUERIES.  35 

between  the  assigned  limits,  or  /*  =  ty,  as  required  by  the 
principle  of  Art.  1 1  to  express  the  probability  of  an  error  lying 
between  those  limits. 


37.   Problems  and  Qtieries. 

1.  Can  cases  be  imagined  where  positive  and  negative  errors  are  not 
equally  probable  ? 

2.  An  angle  is  measured  to  tenths  of  seconds  by  two  observers,  and 
the  value  of  ]i  for  the  first  observer  is  double  that  for  the  second.  Draw 
the  two  curves  of  probability  of  error. 

3.  Show  that  the  arithmetical  mean  of  two  measurements  is  the  only 
value  that  can  be  logically  chosen  to  represent  the  quantity. 

4.  The  reciprocal  of  h  for  the  bullet-marks  in  Art.  18  is  2.33  feet. 
Compare  the  actual  distribution  of  errors  with  the  theoretical. 

5.  Draw  a  curve  for  each  of  the  equations  ji'  =  ke~^'  and_>'  =  /^r~■*-»^^ 
assuming  a  convenient  value  for  k.  Show  that  the  value  of  k  should 
have  been  taken  different  in  the  two  equations. 

6.  Explain  how  the  value  of  tt  might  be  determined  by  experiments 
with  the  help  of  equation  (2). 


36  THE   ADJUSTMENT  OF  OBSERVATIONS.  II L 


CHAPTER   III. 

THE   ADJUSTMENT   OF   OBSERVATIONS. 

38.  The  Method  of  Least  Squares  comprises  two  tolerably 
distinct  divisions.  The  first  is  the  adjustment  of  observations, 
or  the  determination  of  the  most  probable  values  of  observed 
quantities.  The  second  is  the  investigation  of  the  precision  of 
the  observations  and  of  the  adjusted  results.  This  chapter 
contains  the  development  of  the  rules  and  methods  relating  to 
the  first  division. 

Weights  of  Observations. 

39.  Weights  are  numbers  expressing  the  relative  practical 
worth  or  value  of  observations.  Thus,  suppose  a  line  to  be 
measured  twenty  times  with  the  same  chain,  ten  measurements 
giving  934.2  feet,  eight  giving  934.0  feet,  and  two  giving  934.6 
feet  ;  then  the  numbers  10,  8,  and  2  are  the  weights  of  the 
respective  observations  934.2,  934.0,  and  934.6  feet.  Or,  since 
weights  express  only  relative  worth,  the  numbers  5,  4,  and  i,  or 
any  other  numbers  proportional  to  10,  8,  and  2,  may  be  taken 
as  the  weights.  The  observation  934.2  has  cost  five  times  as 
much  as  the  observation  934.6,  and  for  combination  with  other 
measurements  it  should  be  worth  five  times  as  much. 

The  weight  of  an  observation  expresses  the  number  of  stand- 
ard observations  of  which  it  is  the  equivalent.  Thus  the  aver- 
age  of  n  equally  good  direct  measurements  has  a  weight  of  n 


§40-  WEIGHTS   OF  OBSERVATIONS.  3/ 

the  weight  of  each  single  measurement  being  unity.  And 
any  observation  having  a  weight  of  /  *  may  be  regarded  as  the 
equivalent  of  /  observations  of  the  weight  unity,  and  as  having 
a  practical  worth  or  value/  times  that  of  a  single  one.  Hence 
the  use  of  weights  may  be  considered  as  a  convenient  method 
of  abbreviation.  Thus  "934.2  with  a  weight  of  10"  expresses 
the  same  as  the  number  934.2  written  down  ten  times,  and 
regarded  each  time  as  a  single  observation. 

40.  A  weighted  observation  is  an  observation  multiplied  by 
its  weight.  Thus  if  J/„  M^  .  .  .  M„  represent  observations,  and 
/■,  A  •  •  •  /«  their  respective  weights,  the  products  /,J/,,  /2^-^2 
.  .  .  p„M,i  represent  weighted  observations.  If  ,r,,  x^  .  .  .  x„  are 
the  errors  corresponding  to  M„  M^  .  .  .  M„,  the  products  p^x^, 
p2^2.  ■  •  ■  pn^n  may  be  called  weighted  errors.  As  an  error  x  is 
the  difference  between  the  true  and  measured  value  of  the 
quantity  observed,  the  product  px  cannot  occur  without  implying 
that  the  corresponding  observation  M  has  a  weight  of  /,  and 
the  same  is  true  for  the  residual  error  v.  Thus  if  there  be  two 
unknown  quantities  r,^  and  s^,  and  a  measurement  M  be  made 
upon  /(-c,,  ^2),  the  residual  error  is 

if  z^  and  ^^  denote  the  most  probable  values  of  the  unknown 
quantities.  Now,  if  the  observation  i^/be  weighted  with  /,  the 
residual  is 

pv  =  p.f{z,,z^)  -pM. 

Hence  a  v/eighted  observation  always  implies  a  weighted  re- 
sidual, and  vice  versa. 

The  weights  should  be  carefully  distinguished  from  the  meas- 
ures of  precision   introduced  in  the  last  chapter.     The  former 

*  ^  is  the  initial  of  "  pondus." 


38  THE  ADJUSTMENT  OF  OBSERVATIONS.  Ill, 

are  relative  abstract  numbers,  usually  so  selected  as  to  be  free 
from  fractions,  while  the  latter  are  absolute  quantities.  The 
relation  between  them  will  be  shown  in  Art,  43. 


TJie  Principle  of  Least  Squares. 

41.  The  principle  from  which  the  term  "Least  Squares" 
arises  is  the  following  : 

In  measurements  of  equal  precision  the  most  probable  values 
of  observed  quantities  are  those  that  render  the  sum  of  the 
squares  of  the  residual  errors  a  minimum. 

To  prove  this,  consider  the  general  case  of  indirect  observa- 
tions, and  let  ;/  equally  good  measurements  be  made  upon  func- 
tions of  two  unknown  quantities  xr,  and  z-..  Let  J/„  M^  .  .  .  M„ 
be  the  results  of  the  measurements  of  the  functions  /,(-„  -2), 
/,(5„  z^  .  .  .f„{z,,  z^.  These  measurements  will  not  give  ex- 
actly the  true  values  of  the  functions,  and  the  difference  between 
the  observed  and  true  values  will  be  small  errors,  x,,  x^ .  .  .  x„,  or 

/,{z„z,)  -M,  =  x„    Mz„z,)  -M,  =  x,..  .Mz„z,)  -M„  =  x„. 

The  respective  probabilities  of  these  errors  are  by  the  fun- 
damental law  (i) 

_)-,  =  ke-  ''^^^%    y,  =  ke-^'"^^''  .  .  .  y„  =  ke-'"'^n, 

h  being  the  same  in  all,  since  the  observations  are  of  equal 
precision.  Now,  by  Art.  12,  the  probability  of  the  compound 
event  of  committing  the  system  of  independent  errors  ^„  x^ 
.  .  .  x„  is  the  product  of  these  separate  probabilities,  or 

Each  of  these  errors  is  a  function  of  the  quantities  z,  and  z^y 
vhich  are  to  be  determined.     Different  values  of  z,  and  z^  will 


§  4-.  THE    PRINCIPLE    OF  LEAST  SQUARES.  39 

give  different  values  for  P' .  The  most  probable  system  of 
errors  will  be  that  for  which  P' is  a  maximum  (Art.  13),  and 
the  most  probable  values  of  z^  and  z^  will  correspond  to  the 
most  probable  system  of  errors.  The  probability  P'  will  be  a 
maximum  when  the  exponent  of  e  is  a  maximum  ;  that  is  when 

xc  +  x^-  +  x^^  +  .  .  .  +  x,t^  =  a  minimum. 

Hence  the  most  probable  system  of  values  for  z^  and  z^  is  that 
which  renders  the  sum  x,-  +  a\^  +  ,1-3=  -f-  .  .  .  -|-  ,i-„'  a  minimum, 
and  the  fundamental  principle  of  Least  Squares  is  thus  proved. 

The  errors  x„  x^  .  .  .  x„  have  been  thus  far  regarded  as  the 
true  errors  of  the  observations.  As  soon,  however,  as  they  are 
required  to  satisfy  the  condition  that  the  sum  of  their  squares 
is  a  minimum,  they  become  residual  errors  (Art.  8),  so  that  the 
condition  for  the  most  probable  values  of  z^  and  z^  is  really 

(5)  ^'1^  +  ^'2'  +  z^3^  +  .  .  .  "f  v,r  =  a  minimum ; 

that  is  to  say,  if  z,  and  z^  be  the  most  probable  values,  the  com- 
puted residuals 

will  be  those  that  satisfy  the  condition  for  a  minimum. 

The  above  reasoning  evidently  applies  to  any  number  of 
unknown  quantities  as  well  as  to  two. 

42.  The  more  general  case  of  the  Method  of  Least  Squares, 
however,  is  that  when  the  observations  have  different  degrees 
of  precision,  or  different  weights.  In  that  event  the  general 
principle  is  the  following  :  — 

In  measurements  of  unequal  weight  the  m»ost  probable  values 
of  observed  quantities  are  those  that  render  the  sum  of  the 
weighted  squares  of  the  residual  errors  a  niinimuai. 


40  7VIE   ADJUSTMENT  OF  OBSERVATIONS.  III. 

As  before,  let  n  observations,  M^,  M^  .  .  .  M,„  be  made  upon 
functions  of  two  unknown  quantities,  ^,  and  ^^ ;  and  let  /,, 
/z  .  .  .  /«  be  the  respective  weights  of  J/,,  i\L  .  .  .  Jl/„.  The 
differences  between  the  observations  and  the  true  values  of 
the  functions  are  errors,  x„  x\  .  .  .  a\ ;  and  the  respective 
probabilities  of   these  errors  are 

in  which  k  and  //  are  different  for  each  observation.  The  prob- 
ability of  the  system  of  independent  errors,  x„  x^  .  .  .  x„,  then,  is 

and  the  most  probable  system  of  values  is  that  for  which  F'  is 
a  maximum,  or  that  which  renders 

///jc,^  +  h^-X:^  +  .  .  .  +  JinX,^  =  a  minimum. 

The  values  of  ,r,,  x^  .  .  ,  x„,  derived  from  this  condition,  are  the 
residual  errors,  t',,  7'^  .  .  .  •:'„ ;  so  that  it  will  be  well  to  write  at 
once 

/ii^z\^  4-  /h^^'z'  +  .  .  .  +  JinVu   =  a  minimum. 

This  expression  may  be  divided  by  //^  /  /-  being  a  constant 
standard  measure  of    precision  so  selected,  that 

where  pi,  P^  •  •  •  p,,  'ire  whole  numbers,  which  are  the  weights 
of  the  observations  AI^,  M^  .  .  .  M„.*     Then  it  becomes 

(6)  p,v,^  -\-  pzT-'z   4-  .  .  •  -\-  P>iVu   =  a  minimum  ; 


*  To  show  that  these  numbers  are  the  weights  of  M-^,  M^  .  .  .  Af„,  consider  that 
the  condition  for  the  minimum  will  be  fulfilled  when 

dv-i  dv^  dv„ 


§  4-4-  DIRECT  OBSERVATIONS.  4I 

which  is  the  priiiciplc  that  was  to  be  proved.  The  term 
"weighted  square"  means  simply  v~  multiplied  by  the  weight 
/,  or  the  product  pv^. 

The  conditions  expressed  by  {5)  and  (6)  are  the  fundamental 
ones  for  the  establishment  of  the  practical  rules  for  the  adjust- 
ment of  independent  observations.  If  the  observations  are  of 
equal  weight,  the  general  condition  (6)  reduces  to  the  special 
one  (5). 

43.  It  is  here  seen  that  the  squares  of  the  measures  of  pre- 
cision of  observations  are  proportional  to  the  weights,  or  that 

(7)  /'.'  :/'2^  :  J^'  ■■•■p.  :/2  :  A 

The  measure  of  precision  is  never  used  in  the  practical  ap- 
plication of  the  Method  of  Least  Squares,  while  weights  are 
constantly  employed.  The  quantity  //,  however,  is  very  con- 
venient in  the  theoretical  discussions,  and  will  be  needed  often 
in  the  next  chapter :  Ji  represents  an  absolute  quantity,  while  / 
denotes  always  an  abstract  number. 


Direct  Obscfvations  on  a  Single  Quantity. 

44.  When  the  observations  are  of  equal  precision,  and  made 
directly  on  the  quantity  whose  value  is  sought,  it  is  universally 
recognized   that   the   arithmetical   mean   is  the   most   probable 

which,  after  divitlhig  by  the  standard  h'^,  become 

""I  "■'I  "-I 

dv^  dv^  dVfi 

/•"^''  7;,  +  ^-^'3  ;/=;  +  •  •  •  +  /«^'«  ^  =  °- 

Here  the  residual  -'i  is  repeated /i  times,  ~l>^  is  repeated /z  times,  and  v^  is  repeated 
/„  times,  and  hence  p-^,,  f-^  •  •  ■/«  are  the  weights  of  the  corresponding  observations 
M,,  J/,  .  .  .  J/„  (Art.  40). 


42  THE   ADJUSTMENT  OF  OBSERVATIONS.  Hi. 

value  of  the  quantity.     This  may  be  also  shown  from  the  funda- 
mental principle  of  Least  Squares  in  the  following  manner : 

Let  J/,,  J/2  .  .  .  jM„  denote  the  direct  observations  which  are 
all  of  equal  weight  or  precision.  Let  z  be  the  most  probable 
value  which  is  to  be  determined.     Then  the  residual  errors  are 

z  —  J/,,     Z  —  M^  .  .  .  Z  —  Mn, 

and  from  the  fundamental  principle  (5) 

(z  —  My  +  (2  —  My  +  .  .  .  +  (s  —  M,y  =  a  minimum. 

To  apply  the  usual  method  for  maxima  and  minima,  place  the 
first  derivative  of  this  expression  equal  to  zero,  thus 

2(2  -  J/,)  +  2(s  -  J/3)  +  .  .  .  +  2(3  -  Mn)  =  o. 
Dividing  this  by  2,  and  solving  for  ^,  gives 

(8)  0  = 


J/,  +  J/2  +  J/3  +  •  •  •  +  J/« 


n 


that  is,  the  most  probable  value  2  is  the  arithmetical  mean  of 
the  71  observations. 

The  adjustment  of  direct  observations  of  equal  weight  on 
the  same  quantity  is  hence  effected  by  taking  the  arithmetical 
mean  of  the  observations. 

45.  When  the  measurements  of  a  quantity  are  of  unequal 
weight  or  precision,  the  arithmetical  mean  does  not  apply. 
Here  the  more  general  principle  (6)  will  furnish  the  proper  rule 
to  employ.  Let  the  measurements  be  J/„  J/^  .  .  .  J/„,  having 
the  weights/,,  /^  .  .  .  /„.  Then,  if  s  be  the  most  probable  value 
of  the  observed  quantity,  the  expression  (6)  becomes 

/.(2  —  J/i)''  +^2(2  —  J/2)^  +  •  .  .  +A(2  —  J/«)^  =  a  minimum. 


§46.  OBSERVATIOXS   OF  EQUAL    WEIGHT.  ^3 

Placing  the  first  derivative  of  this  equal  to  zero  gives 

/,(2  -  M,)  +M"-  M,)  +  .  .  .  +p,^(^z-Mn)  =  o, 

the  solution  of  which  is 

/„x                .      /, J/,  +  />,M,  +  .  .  .  +p„M„ 
\9)  ^  = ,* 

/.     +  /3     +     .     .     .      +  /„ 

that  is,  the  most  probable  value  of  the  unknown  quantity  s  is 
obtained  by  multiplying  each  observation  by  its  weight,  and 
dividing  the  sum  of  the  products  by  the  sum  of  the  weights. 
In  order  to  distinguish  this  process  from  that  of  the  arithmeti- 
cal mean,  it  is  sometimes  called  the  general  mean,  or  the 
weighted  mean. 

Granting  that  the  arithmetical  mean  gives  the  most  probable 
value  for  observations  of  equal  weight,  the  general  mean  (9) 
for  observations  of  unequal  weight  may  be  readily  deduced  from 
the  definitions  of  the  word  "weight  "  in  Art.  39. 

The  adjustment  of  direct  observations  of  unequal  weight  on 
the  same  quantity  is  hence  effected  by  taking  the  general  mean 
of  the  observations. 

Independent  Observations  of  Equal  Weight. 

46.  The  general  case  of  independent  observations  comprises 
several  unknown  quantities  whose  values  are  to  be  determined 
from  either  direct  or  indirect  measurements  made  upon  them. 

An  "observation  equation"  is  an  equation  connecting  the 
observation  with  the  quantities  sought.  Thus,  if  M  be  a  meas- 
urement of  /(,:-„  .:;,),  the  equation  M  =f{z„  z^  is  an  observation 
equation.  The  number  of  these  equations  is  the  same  as  the 
number  of  observations,  and  generally  greater  than  the  number 
of  unknown  quantities  to  be  determined.     Hence,  in  general, 


44  THE   ADJUSTMENT  OF  OBSERVATIONS.  Ill, 

no  system  of  values  can  be  found  which  will  exactly  satisfy  the 
observation  equations.  They  may,  however,  be  approximately 
satisfied  by  many  systems  of  values ;  and  the  problem  is  to  deter- 
mine that  system  which  is  the  most  probable,  or  which  has  the 
maximum  probability  (Art.  13). 

0  Figf.5.  To  illustrate,  consider  the  following  practi- 
i  "  ^'"■"""---..  z,  cal  case.  Let  O  represent  a  given  bench- 
?                ..,.-''";        mark,    and    Z^,    Z,,   Z^,    three    points    whose 

1  ..-2'"  ?  elevations  above  O  are  to  be  determined. 
^ -4--.         /  Let    five    lines    of    levels    be    run    between 

23         these  points,  giving  the  following  results  : 

Observation  i.  Z,  above  6^  =  10  feet. 

Observation  2.  Z^  above  Z,  =  7  feet. 

Observation  3.  Z^  above  6>  =  18  feet. 

Observation  4.  Z,  above  Z3  =  9  feet. 

Observation  5.  Z^  below  Z,  =  2  feet. 

If  the  elevations  of  the  points  Z.,  Z^,  and  Z3,  be  designated 

by  z^  z^,  and  z^,  the  following  observation   equations  may  be 

written  : 

2,  =  10, 

^2    =      18, 

22  —  23   =       9» 
S.  -  23  =      2, 

each  one  of  which  is  an  approximation  to  the  truth,  but  all  of 
which  cannot  be  correct.  The  number  of  these  equations  is 
five,  the  number  of  the  unknown  quantities  is  three ;  and  hence 
an  exact  solution  cannot  be  made.  The  problem  is  to  find  the 
most  probable  values  of  z^,  z^,  and  ^3. 

The  observation  equations  may  be  algebraic  expressions  of 
the  first,  second,  or  higher  degrees  ;  or  they  may  contain  circular 
or  logarithmic  functions.  Usually,  however,  they  are  of  the 
first  degree,  or  linear,  and  these  alone  will  be  considered  in  the 


^47-  OBSERVATIONS  OF  EQUAL    WEIGHT.  45 

body  of  this  work.  In  Art.  Tzf©  is  given  a  method  by  which 
non-linear  equations,  should  they  occur,  may  always  be  reduced 
to  linear  ones. 

47.  Consider  first  the  case  of  observations  of  equal  precision 
or  of  equal  weight.  Let  there  be  q  unknown  quantities  z^, 
Z2 .  .  .  Zq,  and  let  the  equations  between  them  and  the  xneasured 
quantities  be  of  the  form 

az,  +  dz^  +  .  .  .  +  Izg  =  M, 

in  which  a,  b  .  .  .  I  are  constants  given  by  theory  and  absolutely 
known,  and  J/  the  measured  quantity.  For  each  observation, 
there  will  be  a  similar  equation,  and,  in  all,  the  following  ;/ 
approximate  observation  equations  : 


^3^1    +  ^3^2    + 


dtfli  4-  bnZ^  + 


-f  hz,  =  M„ 

+  kZg  =  M„ 

+  ^3%   =   ^^3. 


~r  '«^^   —   -'";/, 


the  first  of  which  arises  from  the  first  observation,  the  second 
from  the  second,  and  the  last  from  the  ;/'K 

Now,  as  the  number  of  these  observation  equations  is  greater 
than  that  of  the  unknown  quantities,  they  will  not  be  exactly 
satisfied  for  any  system  of  values  that  may  be  deduced.  The 
best  that  can  be  done  is  to  find,  from  the  fundamental  principle 
of  Least  Squares,  the  most  probable  system.  Let  z„  z^  .  .  .  z^ 
denote  the  most  probable  values,  then,  if  these  be  substituted 
in  the  observation  equations,  they  will  not  reduce  exactly  to 
zero,  but  leave  small  residuals,  7',,  v^  .  .  .  2'„]  thus  strictly 

a,z,  +  b,z.  -f-  .  .  .  +  /,c^  —  M,  =  v„ 
a^z,  +  b^z.  +  .  .  .  +  /,s^  —  M^  =  V2, 


dtfii  +  bnZ2  +  •   .   •  +  4%  —  Mn  =  V„. 


46  THE  ADJUSTMENT  OF  OBSERVATIONS.  III. 

The  fundamental  principle  established  in  Art.  41   is,  that  the 
most  probable  values,  ^„  s^  .  .  .  s^,  are  those  that  render 

Vi^  +  V2^  +  ^3^  +  .  .  .  +  v,,^  —  a  minimum. 

Consider  first  what  is  the  most  probable  value  of  the  un- 
known quantity  s„  and  denote  the  terms  in  the  above  equa- 
tions independent  of  ::,  by  the  letters  yV„  A^,  A3,  etc.  Then 
they  become 

a,Zi  +  N,  =  Vi, 
a^Zi  +  A^2  =  V2, 


a„z,  +  N„  =  v„. 


Squaring  both  terms  of  each  of  these  equations,  and  adding  the 
results,  gives 

(a,z,  -f  A^0^+  («.s.+  A^2)^+  . . .  +  (««s.  +  N„y  =  z\'  -f  z',^  +  . . .  +  ^'„^ 

In  order  to  make  this  sum  a  minimum,  its  first  derivative  must 
be  put  equal  to  zero,  giving 

c.(a.s,  +  JV,)  +  a^ia^z,  +  A^)  +  .  .  .  +  a„{a„z,  +  N„)  =  o  ; 

and  this  is  the  condition  for  the  most  probable  value  of  s^.  In 
like  manner  a  similar  condition  may  be  found  for  each  of  the 
other  unknown  quantities.  The  number  of  these  conditions, 
or  "normal  equations"  as  they  are  called,  will  be  the  same  as 
that  of  the  unknown  quantities,  and  their  solution  will  furnish 
the  most  probable  values  of  z^,  z^  .  .  .  z^. 

48.  The  following  is,  hence,  the  method  for  the  adjustment 
of  independent  indirect  observations  of  equal  weight : 

For  each  observation  write  an  observation  equation.  Form 
a  normal  equation  for  z,  by  multiplying  each  of  the  observation 
equations  by  the  co-efificient  of  z,  in  that  equation,  and  adding 


§48.  OBSERVATIONS   OF  EQUAL    WEIGHT.  47 

the  results.  And,  for  each  unknown  quantity,  form  a  normal 
equation  by  multiplying  each  observation  equation  by  the  co- 
efficient of  that  unknown  quantity  in  that  equation,  and  adding 
the  results.  The  solution  of  these  normal  equations  will  fur- 
nish  the  most  probable  values  of  the  unknown  quantities. 

For    example,  let    the    five    observation    equations    derived 
from  the  five  observations  of  Art.  46  be  considered,  namely, 


^I 

' — 

10, 

z. 

+ 

Z2 

= 

7, 

Z2 

= 

18, 

^2 

-^3 

— 

9. 

z. 

-  ^3 

= 

2. 

To  form  the  normal  equation  for  2,  the  first  observation 
equation  is  multiplied  by  -f-  i,  the  second  by  —  i,  the  third  by 
O,  the  fourth  by  o,  and  the  fifth  by  -f  i  ;  the  addition  of  the 
products  then  gives 


301    —  2,   —  2^3   =  5. 

The  normal  equation  for  s^  is  formed  by  multiplying  the  first 
observation  equation  by  o,  the  second  by  -|-  i,  the  third  by  -{-  i. 
the  fourth  by  -|-  i,  and  the  fifth  by  o ;  the  sum  of  the  products 
being 

—  z,  -\-  3^2  —  %  =  34. 

The  normal  equation  for  ^3  is  formed  by  multiplying  the  first, 
second,  and  third  observation  equations  by  O,  and  the  fourth 
and  fifth  by  —  i,  the  addition  of  which  gives 

—  ^i  —  ^2  +  2S3  =  —  1 1. 

These  three  normal  equations  contain  three  unknown  quanti- 
ties, and  their  solution  gives 

s,  =  +  io|,     z,  =  -\-  i7§,     23  =  4-  8^, 


48  THE   ADJUSTMENT  OF  OBSERVATIONS.  III. 

which  are  the  most  probable  values  that  can  be  obtained  from 
the  five  observations.  If  now  these  values  be  substituted  in 
the  observation  equations  there  will  be  found  the  five  residuals, 


—  1 

8' 


z;,  =  4-1,     t'.  =  +i,     v^=-l,     v,  =  ^\,     v^  = 

and  the  sum  of  the  squares  of  these  is  |.  Of  all  the  possible 
values  that  might  be  assigned  to  ^„  s^,  z^  those  above  found 
give  the  minimum  sum  of  squares  of  residual  errors. 

As   a  second   example,   let    three  observations  on   the   two 
quantities  z^  and  z^  give  the  observation  equations 

3^1  —  5^2  =  +  ^2-4, 

—  2Zi   -\-  4G2   =    —   10.2, 

Si    —    202   =^   ~H      8.0. 

To  form  the  normal  equation  for  s,  the  observation  equations 
are  multiplied  by  4-3.-2,  and  +  i,  respectively,  and  the  re- 
suits  added.  To  form  the  normal  equation  for  z^  the  multipliers 
are  —  5,  +4,  and  —  2,  respectively.  The  two  norma)  equa- 
tions thus  are 

14S1  —  2502  =  +    65.6, 

—  25s,  +  45^2  =  —  118. 8, 

and  the  solution  of  these  gives  the  most  probable  values 
z^=z  —  3.60  and  z^^  —  4.64. 

49.  In  order  to  put  the  above  method  for  the  formation  of 
normal  equations  into  algebraic  language,  let  there  be  n  obser- 
vations  upon  q  unknown  quantities  which  lead  to  the  following 
observation  equations : 

aiZi  -f  biz^  +  c,z-i  -\- . . .  +  hzg  =  J/i, 

/      s                   a2Zi  +  b2S2  +  ^2^3  +  . . .  +  /2-?  =  ^-^2, 
(10)  

tZ„Si  +  bnZ2  +  ^«-3  -\-  '  •  '   +  ^nZq  —  Mn. 


§49-  OBSERVATIONS   OF  EQUAL    WEIGHT.  49 

The  normal  equation  for  z^  is  formed  by  multiplying  the  first 
of  these  by  a^,  the  second  by  a^,  the  last  by  ^„,  and  adding  the 
products,  thus  giving 

{a^  +  ^2'  +  •  • .  +  an)^x  +  {fl^bi  +  aj)^  +  . . .  +  a,,/^,,)^^  +  .  . . 

and  in  like  manner  a  normal  equation  for  each  of  the  other 
unknown  quantities  may  be  written.  To  simplify  the  expres- 
sion of  these  equations,  let  the  following  abbreviations  for 
summation  be  introduced  : 

\aa\     —  a,^       +  a.'      +  it^      +  .  .  .  +  a„^ 
\ab\     —  a^b,     +  ^21^2     +  a^bi     +  •  •  •  +  dubn, 
[a/j      =  aji      -\-  a^h     -f-  aji     -)-...  -j-  ajn, 
\bb\      =  bf       ^bf      -\-b.;-       +  .  . .  +  b,?, 
[aM]  =  a, Ml  +  ajl2  -j-  ajf^  +  . .  .  -\-  a„M„, 

and  then  the  normal  equations  may  be  thus  written: 

[aa]z,  +  [ab]-.  +  [ac]c^  +  •  •  •  +  M^,  =  [a^], 
[ba]z,  +  [bb],.  +  [bc]z,  +...  +  \b/]2,  -  [bA/], 
(11)    [ca]o,   4-  [cb]c,  +  [cc]z,  +...  +  [c/]^,  =  [cM], 

\la\z,   +  \lh\z.  +  \_lc\z,  +  . . .  +  [//]c,  =  [/J/]. 

The  co-ef^cients  of  the  unknown  quantities  in  these  normal 
equations  present  a  curious  symmetry  ;  those  of  the  first  hori- 
zontal row  being  the  same  as  those  of  the  first  vertical  column, 
those  of  the  second  row  the  same  as  those  of  the  second  col- 
umn, and  so  on.  This  is  due  to  the  fact  that  \ba\  is  the  same 
as  \ab\  \ca\  the  same  as  \ac\  . . .  and  [/^]  the  same  as  [a/]. 

The  notation  for  summation  here  indicated  is  that  first  used 
by  Gauss  and  since  generally  employed  in  works  on  the  Method 
of  Least  Squares  in  writing  normal  equations.  The  notation 
2a\  2ai?,  used  by  a  few  writers,  and  in  former  editions  of  this 


50  THE  ADJUSTMENT  OF  OBSERVATIONS.  HI. 

book,  has  the  same  meaning  as  \aa\  \ab'\.  The  sum  of  the 
squares  of  the  residual  errors  may  be  written  either  ^v''  or 
[z-'Z'],  and  in  this  book  the  former  will  be  employed  as  it  more 
readily  calls  to  mind  its  name. 

50.  Hence  the  method  of  adjustment  of  indirect  observa- 
tions of  equal  weight  is  to  write  for  the  ;/  observations  the  n 
observation  equations  (10),  then  to  form  the  ^  normal  equations 
(11),  and  their  solution  will  furnish  the  most  probable  values 
of  the  unknown  quantities.  Numerous  examples  of  the  appli- 
cation of  this  method  will  be  found  in  Chap.  VII. 

As  a  simple  illustration  let  three  observation  equations  be 

431  —   2Z,  ■=   +6.1, 
53,  +   23,  =   +  3.8, 

33i  —  Z^2—  —  0.9. 

Here  a,  —  +  4,  a,  =  -f  5,  ^3  =  +  3,  <^,  =  —  2,  /^,  =  +  2, 
^3  =  —  3,  M,  =  +  6.1,  M^  =  +  3.8,  M^  =  -  0.9.  The  forma- 
tion of  the  sums  is  now  made,  carefully  regarding  the  signs  of 
the  co-efficients ;  thus, 

[,ia]  =  +  4^  +  5=  +  3^  z=  +  50.0, 
[a/?]  —  —  8  +  10  —  9  =—  7.0, 
[«J/]  =  +  24.4  +  19.0  -  2.7  =  4-  40.7, 
[bb^      =  2^+       2^+    3'=  +  i7-0' 

{bM]  =  —  12.2  +  7.6    +  2.7  =  —     1.9. 

Here  [^ba]  need  not  be  computed,  as  its  value  is  the  same  as 
[^ab];  thus  the  two  normal  equations  are 

+  502,  —    7^2  =  +  40-7. 
—    7S1  +  17S2  =  —    1.9, 

the  solution  of  which  gives  z,  =  +  0.8472  and  z^  =  4-  O.2371  as 
the  most  probable  values  correct  to  the  fourth  decimal  place. 


§52.  OBSERVATIONS   OF   UNEQUAL    WEIGHT.  5 1 


Independent  Observations  of  Unequal  Weight. 

51.  The  more  usual  case  in  practice  is  where  the  observa- 
tions have  unequal  weights.  As  weights  are  merely  numbers 
denoting  repetition,  it  is  plain  that  if  each  observation  equa- 
tion be  written  as  many  times  as  indicated  by  its  weight,  the 
reasoning  of  Art.  47  and  the  rule  of  Art.  48  applies  directly 
to  the  determination  of  the  probable  values  of  the  unknown 
quantities.  Instead,  however,  of  writing  an  observation  equa- 
tion as  many  times  as  indicated  by  its  weight,  it  will  be  sufificient 
to  multiply  it  by  its  weight  when  forming  the  other  products. 

52.  The  following  rule  may  hence  be  stated  for  the  adjust- 
ment of  independent  observations  of  unequal  weight  upon 
several  related  quantities: 

For  each  weighted  observation  write  an  observation  equa- 
tion, noting  its  weight.  Form  a  normal  equation  for  z,  by 
multiplying  each  equation  by  the  co-efificient  of  z^  in  that  equa- 
tion, and  also  by  its  weight,  and  adding  the  products.  In  like 
manner  form  a  normal  equation  for  each  of  the  other  unknown 
quantities  by  multiplying  each  observation  equation  by  the  co- 
efificient  of  that  unknown  quantity  in  that  equation,  and  also 
by  its  weight,  and  adding  the  results.  The  solution  of  these 
normal  equations  will  furnish  the  most  probable  values  of  the 
unknown  quantities. 

For  example,   let    three   observations    upon    two    unknown 
quantities  give  the  three  observation  equations, 

—  2Z,  4-  33^  =  -f  6,     weight  3, 

+  20,  =  +  3,     weight  7, 

—  3^2  =  +  5>     weight  2. 

To  form  the  normal  equation  for  ^,  the  first  equation  is  multi- 
plied by  the  co-efficient  —  2  and  by  the  weight  3,  that  is,  by 


$2  THE  ADJUSTMENT  OF  OBSERVATIONS.  III. 

—  6;  the  second  is  multiplied  by  -|-  2  and  7,  that  is,  by  -f-  Hi 
the  third  is  multiplied  by  o  and  2,  that  is,  by  o;  the  addition 
of  the  products  gives 

-f  |oSi  —  18^2  ==  +6. 

To  form  the  normal  equation  for  s^,,  the  first  equation  is  mul- 
tiplied by  -|-  3  and  by  3,  that  is,  -]-  9  ;  the  second  by  o,  and  the 
third  by  —  6;  the  sum  of  products  being 

—  i8si  -f  45S2  =  +  24. 

The  solution  of  these  two  normal  equations  gives  s^^^  4'  0.475 
and  ^^2  =  +  0.724  as  the  most  probable  values  of  the  two 
quantities  which  were  indirectly  observed. 

53.  In  order  to  put  this  method  into  an  algebraic  algorithm 
and  at  the  same  time  review  the  general  reasoning,  let  M^, 
M2,  . .  .  M„  be  the  results  of  the  ;/  observations  which  have  been 
made  to  determine  the  values  of  the  g  quantities  z„  s^,  . .  .  z^. 
As  before,  let  each  observation  be  represented  by  an  observa- 
tion  equation,  thus  : 


(12) 


a,z,  -f  biZ.  +  . . .  +  hzq  =  J/i     with  weight /i, 
a2Z,  +  ^202  +  ...4-/2^^  =  ^I-     with  weight /2, 

On"-!  +  i>nZ2-\-  .  .  .  +  hiZq  =  Mn     witli  Weight/,,. 


Now,  if  z^,s^...z^  denote  the  most  probable  values  of  the 
quantities  sought,  and  these  values  be  substituted  in  (12),  these 
equations  will  not  reduce  to  zero,  but  leave  small  residuals,  z\^ 
v^  ..  .Vn.     Thus  strictly, 

a^z,  4-  hz2  -y  .  ..Ar  h"q—  M^  -  v,     with  weight  p,, 
a2Zi  4-  l'2Z-2  -\-  ...-{-  hzq  —  M2  —  v-2     with  weiglit  p^. 


a„zi^  bnZ^-^  ...  4  A,s<7  —  Mn  —  v„    with  weight/, 
which  may  be  called  residual  equations. 


m 


§53-  OBSERVATIONS   OF  UNEQUAL    WEIGHT.  53 

Now,  according  to  the  general  principle  (6),  the  most  probable 
values  of  the  q  unknown  quantities  ^,,  s^  .  .  .  z^  are  those  that 
render  the  expression 

piVi^  -\-  P2V2''  +  .  .  .  +  pnVn''  =  a  minimum. 

To  abbreviate,  designate  this  quantity  by  l.pv~.  Rememberino- 
that  7',^  7-2^  .  .  .  v,^  are  functions  of  s„  z^  .  .  .  Zg,  it  is  plain  that 
the  derivative  of  2/7'=  with  reference  to  each  variable  must  be 
zero,  and  that  hence  there  are  the  following  q  conditions  for 
the  minimum  : 

.      d7>j    ,    ^      ^V2  ,  ,    ^      dv„ 

dz,  dz,  dz^ 

.      dv,    ,    ^      dv^    ,  ,    ^       dv„ 

Pi'v,  --   +p2V2 h  .  .  .  +  PhVh  ~—  =  o, 

dz,  dz^  dz. 


dv,    ^    ^      dvj,   ,  ,  dvn 

Pi'Vx f-  p2V2 }-...+  pnVu  3—  =  O. 

dZq  dZq  dZq 

The  values  of  the  differential  co-efficients  in  these  conditions 
are  readily  found  by  taking  the  derivatives  of  the  residual 
equations  with  reference  to  each  variable,  thus  : 

dt\  dv2  dz\       , 

- —  =  a,,     —  =  a2,     —  =  Oi,  etc. ; 

dZi  dzi  dz^ 

and  the  conditions  then  become 

PxdyV^   +  piC.V.   4-   .   .   .  +  p„anV„  =  O, 
Pi^iV,    +  p2l>2V2    +   .   .   .   +  pnbnVn    =   O, 

pJ.Vx    -f-  p2hV2    +    .   .   .    -f  pJnVn    =  O, 

which  are  as  many  in  number  as  there  are  unknown  quantities 
-s-,,  Z2  .  .  .  Zg.     If  in  these  the  values  of  v^,  z'^  .  .  .  v„  be  replaced 


54  THE  ADJUSTMENT   OF  OBSERVATIONS.  HI. 

from  the  residual  equations,  the  final  normal  equations  will 
result.  As  before,  the  expression  of  the  normal  equations  may 
be  abbreviated  by  using  the  square  bracketed  notation  for 
summation,  namely, 

\_pab\       =:  piUxbi     -\- p2(iib2    +  •  •  •  -\-pnanbn, 
\_paAf]  =  p^aMi  -V  P^<^2M2  +  .  .  •  -\-  pnOnM,,,  etc., 

and  thus  the  normal  equations  are 

[paa\z,  +  [pab^z^  +  •  •  •  +  [A^/I^,  =  [paM\ 

.     .          [pba]z,  +  [pbby.2  +  .  .  .  +  [A>/]^,  =  VpbM], 
(13)  

[phi-\z.   +  [plb-\z2  +  .  .  .  +  [///]^^.  =  [//^/], 

by  whose  solution  the  most  probable  values  of  s^,  z^  .  .  .  z^  may 
be  found.  The  co-efficients  in  these  equations  show  the  same 
symmetry  as  those  in  Art.  48,  since,  as  before,  \_pba\  \_plb\ 
etc.,  are  the  same  as  [pab],  [p^/],  etc. 

54.  Thus,  if  there  be  u  observations  for  determining  ^  un- 
known quantities,  the  most  probable  values  of  the  unknown 
quantities  are  obtained  by  writing  n  observation  equations  as 
in  (12),  and  forming  the  q  normal  equations  as  in  (13);  then 
the  solution  of  these  normal  equations  will  furnish  the  most 
probable  values  of  the  g  unknown  quantities.  In  the  most 
common  cases  the  co-efficients  in  the  observation  equations 
(12)  are  -|-  I,  —  I,  or  o,  and,  in  the  formation  of  the  co-effi- 
cients of  the  normal  equations,  the  signs  must  be  carefully 
regarded.  Many  examples  of  adjustment  by  this  method  are 
given  in  Chap.  VII. 

As  a  simple  illustration  let  there  be  given  the  follovving  four 
weighted    observation   equations   upon   the   two   quantities  s, 


ana  z^ : 


§54-  SOLUTION   OF  NORMAL   EQUATIONS.  55 

No.  I.         +2,  =  o,  weight/,  =    8, 

2.  •\-     Z:,  =  o,  /^  =  lo, 

3.  +  5x  +   2C3    =    +  0.25,  /3=       I, 

4-        +  ^i  -  3-^  =  -  0-92,  A  =    5. 

These  co-efficients  and  weights,  arranged  in  tabular  form,  are 


No. 

a 

b 

il/ 

/ 

I. 

+  1 

0 

0 

8 

2. 

0 

+  1 

0 

10 

3- 

+  1 

+  2 

+  0.25 

I 

4- 

+  1 

-3 

—  0.92 

5 

The  products  paa,  pab,  etc.,  are  now  formed  as  below,  and 
their  summation  furnishes  the  co-efficients  for  the  two  norma) 
equations  ;  thus, 


No. 

paa 

pab 

paM' 

pbb 

pbM 

I. 

+    8 

0 

4 

0 

0 

0 

2. 

0 

0 

0 

+   10 

0 

3- 

+     I 

+      2 

+  0.25 

+    4 

+    o.5(> 

4- 

+    5 

-    15 

—  4.60 

+  45 

+  i3-8'> 

+  14  -  13  -  4-35  +59  +  14.30 

Here  \_pbd\  is  the  same  as  \^pab'\  and  need  not  be  computed. 
The  normal  equations  therefore  are 

+  14^1  —  13^2  =  —    4.35, 
—  13^1  +  59^2  =  +  i4-3o> 

the  solution  of  which  gives  2-,  =  —  0.102  and  z^=-  -\-o  225  a? 
the  most  probable  values  that  can  be  derived  from  the  four 
observations.  If  these  be  substituted  in  the  observation  equa- 
tions the  adjusted  values  of  the  four  observations  are  found  to 
be  —  0.102,  -|-  0.225,  -f-  0.348,  and  —  0.777. 


56  THE  ADJUSTMENT  OF  OBSERVATIONS.  III. 

In  the  formation  of  the  co-ef^cients  of  normal  equations 
tables  of  squares,  multiplication  tables,  and  calculating  ma- 
chines will  often  be  found  very  useful.  The  method  of  using 
the  table  of  squares  at  the  end  of  this  volume  for  the  forma- 
tion of  the  products  ab,  ac,  etc.,  is  explained  in  Art.  172,  and 
a  method  for  checking  the  correctness  of  the  co-efificients  \ab\ 
\ac\  etc.,  is  given  in  Art.  142. 

Solution  of  Normal  Equations. 

55.  The  normal  equations  which  arise  in  the  adjustment  of 
observations  may  be  solved  by  any  algebraic  process.  When 
the  co-efificients  consist  of  several  digits,  or  when  the  number 
of  unknowns  is  greater  than  three,  it  is  desirable  to  follow  a 
method  by  which  checks  may  be  constantly  obtained  upon 
the  accuracy  of  the  numerical  work.  Such  a  method,  devised 
by  Gauss,  is  presented  in  Chap.  X. 

When  the  number  of  unknown  quantities  is  two,  the  obser- 
vation equations  furnish  the  two  normal  equations 

\aa\z^  -f-  \ab\z2  —  \aM\ 
\ab-\z,  +  \bby.._  =  \bM\ 

the  solution  of  which  may  be  directly  effected  by  the  formulas 

^  {bb-\\aM\  -  \c,b\bM\ 
laa\  \bb\  -  \abY      ' 

^^-  iaa\bb]-la-t>f'^ 

while  checks  upon  the  numerical  work  may  be  obtained  by 
substituting  the  computed  values  in  the  given  normal  equa- 
tions which  should  be  exactly  or  closely  satisfied. 

When  logarithms  are  used  it  will  generally  be  advantageous 
to  write  the  formulas  thus 


§  54-  CONDITIONED    OBSERVATIONS.  $7 

_  [g^][/;.l/]/[^^]  -  [a A/] 

as  then  the  table  need  be  entered  only  three  times  in  finding 
the  numbers  corresponding  to  two  terms  in  the  numerator  and 
one  in  the  denominator,  whereas  by  the  former  formulas  six 
entries  are  required. 

As  an  example  let  the  two  normal  equations  be 

90.07S1  +    40456^2  =    295.99, 
404.56:^1  +  1934.10^2  =  1306.90. 

Here,  by  the  use  of  either  numbers  or  logarithms,  the  solution 
gives  the  values 

2i  =  +  4.1527,  22  =-0.1929, 

which,  substituted  in  the  normal  equations,  reduce  the  first  to 
+  0.004  =  0  and  the  second  to  +  0.028  =  o.  The  first  is 
satisfied  as  closely  as  the  data  admit,  while  the  error  in  the 
second  can  be  reduced,  if  deemed  necessary,  by  carrying  the 
values  of  z,  and  s^  to  five  decimal  places. 

When  the  number  of  unknown  quantities  is  three,  general 
formulas  for  solution  are  best  derived  in  the  determinant  form 
given  in  Art.  140.  This  determinant  method  is  easily  remem- 
bered and  may  be  advantageously  used  for  the  case  of  two 
unknown  quantities. 

Conditioned  Observations. 

56.  Thus  far  it  has  been  considered  that  the  quantities  to 
be  determined  by  observation  were  independent  of  each  other. 
Although  they  have  been  related  to  each  other  through  the 


58  THE  ADJUSTMENT  OF  OBSERVATIONS.  III. 

observation  equations,  and  have  been  required  to  satisfy  ap- 
proximately those  equations,  they  have  been  so  far  independent, 
that  any  one  unknown  quantity  might  be  supposed  to  vary 
without  affecting  the  values  of  the  others.  All  systems  of  values 
of  the  unknown  quantities  have  been  regarded  equally  possible, 
and  the  methods  above  developed  show  how  to  determine  the 
most  probable  system. 

In  the  class  of  observations  now  to  be  discussed,  all  systems 
of  values  are  not  equally  possible,  owing  to  the  existence  of 
conditions  which  must  be  exactly  satisfied.  Thus,  having 
measured  two  angles  of  a  triangle,  the  adjusted  value  of  one 
is  entirely  independent  of  that  of  the  other ;  but,  if  the  third 
angle  be  measured,  the  three  angles  are  subject  to  the  rigor- 
ous geometrical  condition  that  their  sum  must  be  exactly  i8o°. 
In  conditioned  observations  there  are,  hence,  two  classes  of 
equations,  observation  equations  and  conditional  equations ; 
the  number  of  the  first  being  generally  greater  than  the  number 
of  unknown  quantities,  and  that  of  the  latter  always  less.* 

57.  Designate  the  number  of  observation  equations  by  ;/,  the 
number  of  unknown  quantities  by  q,  and  the  number  of  condi- 
tional equations  by  ;/.  If  no  conditional  equations  existed,  the 
principle  of  Least  Squares  (6)  would  require  that  the  adjusted 
system  of  values  should  be  the  most  probable  for  the  n  inde- 
pendent observations.  The  ;/  conditional  equations,  being  less 
in  number  than  the  q  unknown  quantities,  may  be  satisfied  in 
various  ways;  and,  further,  the  final  adjusted  system  of  values 
must  exactly  satisfy  them.  Hence  it  must  be  concluded,  that, 
of  all  the  systems  of  values  which  exactly  satisfy  the  ;/  condi- 
tional  equations,  that  one   is   to    be    chosen,  which    in    the  n 


*  In  most  books  upon  this  subject,  the  term  "equations  of  condition"  is  applied 
indiscriminately  to  both  of  these  very  distinct  classes,  and  is  a  cause  of  some  per- 
plexity to  the  student.  The  excellent  distinction  of  the  Germans,  "  Beobachtungs- 
gleichung"  and  "  Bedingungsgleichung,"  ought  certainly  to  come  into  use. 


§58.  CONDITIONED   OBSERVATIONS.  59 

observation  equations  makes  the  sum  of  the  weighted  squares 
of  the  residuals  a  minimum. 

The  problem  of  conditioned  observations  may  be,  then, 
reduced  to  that  of  independent  ones  by  finding  from  the  ;/' 
conditional  equations  the  values  of  ;/'  unknown  quantities  in 
terms  of  the  remaining  q  —  ;/  quantities,  and  substituting  them 
in  the  n  observation  equations.  There  will  thus  result  ;/  inde- 
pendent observations  upon  q  —  ;/'  quantities.  From  these  the 
normal  equations  are  to  be  formed,  and  the  most  probable 
values  of  the  q  —  ;/  quantities  deduced.  Substituting  these 
values  in  the  ;/'  conditional  equations,  the  remaining  ;/  quan- 
tities become  known.  Thus  the  q  quantities  exactly  satisfy 
the  conditional  equations,  and  at  the  same  time  are  the  most 
probable  values  for  the  observation  equations.  This,  therefore, 
is  a  general  solution  of  the  problem. 

For  example,  consider  the  measurement  of  the  three  angles 
of  a  plane  triangle.  Let  ^„  z^,  and  z^  be  the  most  probable 
values  of  the  angles,  and  let  the  observation  equations  be 

0,  =  J/,,     0,  =  J/2,     S3  =  J/3, 

which  are  subject  to  the  rigorous  condition 

2,  +  "2  +  S3  =  180°. 

From  the  conditional  equation  take  the  value  of  ^3,  and  substi- 
tute it  in  the  observation  equations,  giving 

Z,      =      J/„  Z2      =      M^,  Zl      +     ^2      =       180°      —     My 

The  most  probable  values  of  z^  and  z^  may  be  now  obtained 
by  the  method  of  Art.  47,  since  the  three  observation  equa- 
tions are  independent.  Then  the  most  probable  value  of  z^  is 
180°  —  z^  —  z^. 

58.  Although  the  above  method  is  perfectly  general,  and  very 
simple  in  theory,  it  gives  rise  in  practice  to  tedious  computa- 


6o  THE   ADJUSTMENT  OF  OBSERVATIONS.  III. 

tions  whenever  the  number  of  conditional  equations  is  large. 
The  process  generally  employed  is  the  "method  of  correlatives," 
due  to  Gauss,  which  will  now  be  explained  ;  the  conditional 
equations  being  considered  as  of  the  first  degree,  or  linear, 
and  the  number  of  observations  being  the  same  as  that  of  the 
quantities  to  be  determined,  or  n  =  q. 

Consider  q  unknown  quantities  connected  by  the  n'  rigorous 
conditions, 

a-o  -\-  a,S,   4-  a^z^  +  .  .  .  +  OqZq  =  O, 
/So  +  /3,3,    +  /S,S3  +  .  .  .  +  ^gZq  =  o, 


(14) 


Ao  +  A,2,  +  A,s,  +  .  .  .  +  \gZq  —  o 


Let  J/„  J/2  .  .  .  Mq  be  the  values  found  by  the  observations  for 
z„  S2  .  .  .  Zg.  If  these  be  inserted  in  the  conditional  equations, 
they  will  not  reduce  to  zero,  but  leave  small  discrepancies, 
</„  dz  .  .  .  dn',  thus  : 

Oo  4-  a,J/,  +  o.zM'^  +  .  .  .  +  aqMg  =  d„ 

/3o  +  (3Jf,    +  /3J/,   +   .    .    .   +  fSqA/q   =    d„ 


Xo  -f-  A,yl/,  +  XJf^  +  .  .  .  +  \Mq  =  d,,'. 

Let  v^,  V2  .  .  .  Vq  be  corrections,  which  when  applied  to  J/„ 
M^  .  .  .  Mg,  wall  render  them  the  most  probable  values,  and 
cause  the  discrepancies  to  disappear ;  thus,  if  s^,  s^  .  .  .  z^  be 
the  most  probable  values, 

2,    =    yl/i   +  V^,        Z2   =   M^   +  V2   .    .    .  Zq   =   Mg  +  Vq. 

Then  the  insertion  of  these  in  (14)  gives  the  reduced  condi- 
tional equations 

a,z;,   +  a-^V^.  +  .  .  .  +  agVq  +  «',  =  O, 

/  y  fi^V^    +    ^iV^    +    .     .     .     +    ^gVg    +    4     =    0, 

XyV^  +  AgJ'z   +  .   .   .  +  \gVq  +  dn'  =  O. 


§  5<^-  CONDITIONED    OBSERVATIONS.  6 1 

For  the  sake  of  shortness,  let  these  ;/  conditions  be  expressed 
by  the  notation  /(a),/(^)  .  .  .  /(A). 

Let  /„  p2  ■  ■  -pg  be  the  weights  of  the  observations  M„ 
M,  .  .  .  Mq.  The  corrections  v„  v,  .  .  .  v^  are  the  same  as  the 
residual  errors,  and  the  sum  of  their  weighted  squares  is  repre- 
sented by  2/z'-.  The  most  probable  values  of  z„  z^  .  .  .  c  are 
those  that  render  a  minimum  the  expression 

%pzr~  -  2AV(a)-   2AV(yS)  -  ...  -  2K„'f{\), 

where  A'„  K^  .  .  .  K,,'  are  multipliers,  or  "correlatives,"  of  the 
conditional  equations.* 

The  derivative  of  this  expression  with  reference  to  each  v  is 
to  be  put  separately  equal  to  zero,  thus  :  — 

p,v,  —  {a,K,  +  ^,K^  +  .  .  .  +  A, A'„')  =  o, 
p^v^  —  {a^K^  +  ^^K^  +  .  .  .  +  X^K„')  —  o, 


pgVg  —  {agK^  -f  PgK^  +  .  .  .  +  X,iK„-)  =  o. 

These  q  equations  together  with  the  //  conditional  equations 
are  suf^cient  for  the  determination  of  the  q  residuals  and  the 
n'  correlatives.     The  residuals  may  be  written 

(^c\  z'2  = -A, -f  ^  A2  + .  .  .  4- — A,/, 


V,  =  "^K,  +  ^^  A'.  +  .  .  .  +  i^  AV, 

A         A  A 


*  It  is  shown  in  works  on  the  differential  calculus,  that  the  maximum  or  mini- 
mum of  a  function,  F{x,y,z),  whose  variables  are  connected  by  conditional  equa- 
tions, 0(jr,  jt',  s)  =  o,  Q[x,y,z)  =  o,  is  to  be  found  by  multiplying  the  conditional 
equations  by  undetermined  co-efficients,  adding  them  to  the  function,  and  then, 


62  THE   ADJUSTMENT  OF  OBSERVATIONS.  III. 

and,  if  these  be  substituted  in  the  reduced  conditional  equa- 
tions, they  become 


=  o. 


[y>-[^ 


^1^ -..  ■  r^i 


/ 


PA 


Kn'  +  dn'  —  O, 


in  which  the  usual  notation  for  sums  is  followed,  for  example : 


^i 


qr^q 


The  co-eflficients  in  these  equations  have  similar  properties 
to  those  in  the  normal  equations  derived  for  independent  ob- 
servations, those  of  the  first  row  being  the  same  as  those  of 
the  first  column,  and  so  on.  Being  n  in  number  they  deter- 
mine the  n'  correlatives;  and  the  residuals  v  are  then  known. 
These  residuals,  applied  as  corrections  to  the  observations, 
give  the  most  probable  values  of  the  quantities  s^,  z^.  .  .  z^,  and 
these  must  exactly  satisfy  the  q  conditional  equations. 

58<z.  As  an  illustrative  example  let  there  be  five  quantities 
connected  by  the  two  conditional  equations, 

-S,  +  ^2  —  ^3  =  o.  ^2  —  ^4  +  2:5  =  o, 

and  let  the  results  of  five  observations  be 

Zi  =  lo.i,     "i  =  6.6,     ^3  =  18.0,     2,  =  9.2,     z,  =  2.7  inches. 


by  the  usual  rule,  determining  the  maximum  or  minimum  of  the  new  function, 
F(x,  y,  z)  +  c<p<x,  y,  z)  +  £'B(x,  y,  z). 

The  minimum  2pv^  only  gives  the  most  probable  values  of  the  unknown 
quantities  when  these  are  independent.     (See  Art.  42.) 


§58^-  CONDITIONED    OBSERVATIONS.  63 

the  weights  of  these  observations  being 

/.  =  1,    A  =  2,    /3  =1,    p^=  I,    p^  -  i^ 
It  is  required  to  adjust  the  observations. 

By  comparing  the  conditional  equations  with  (14)  are  found 
a,  =  o,    a,  =  +  i,    cx,=  +  i,    «'3  =  -  I,    a^  =  0,         a^  =  o, 
/?o-o,    A  =  o,         A=  +  i,    A-o,         /3,=  -i,     /3^  =  -{-i. 

Also  by  substituting  the  observed  values  in  the  conditional 
equations  are  derived  d,  =  —  1.3  and  c/,  =  -f  o.i.  The  co- 
efficients of  the  equations  (16)  are  next  found  ;  for  example 


[ 


7j 


~7~i~:7~'~"""|"  -  +  -  =  +  2.5. 
I      2      I      I      I       >     J 


The  correlative  normal  equations  themselves  then  are 

2.5^,  +  o  5^-3  -  1.3  -  o, 

0.5  A',  +  2.5A'3  +  0.1   r=  o, 

whence  K,  =  +0-55  and  K,  =  -0.15.     From  (15)  the  most 
probable  corrections  to  the  observed  values  are  found  to  be 

^i  =  +o.S5,    ?'2=+o.2o,    z;3  =  — 0.55,    J74  =  -1-0.15,    7'5=— 0.15, 
and  the  final  adjusted  values  are 

2.  =  10.65,     ^2  =  6.80,     ^3=  17.45,     ^4  =  9-35'    ^5  =  2.55, 
which  exactly  satisfy  the  two  conditional  equations. 

This  problem  may  be  reduced  to  one  of  independent  obser- 
vations by  eliminating  two  of  the  unknowns  by  means  of  the 
conditional  equations.  Thus,  if  s^  and  x;^  be  eliminated  the 
observation  equations  are  c,  =  10.  i,  s, -{- ja,  =  18.0,  s^  =  9.2, 
—  s  -\-  z^  =  2.7,  all  with  weight  i,  and  z^  =  6.6  with  weight  2. 


64  THE  ADJUSTMENT  OF  OBSERVATIONS.  Ill 

58^.  A  valuable  check  upon  the  solution  of  the  correlative 
normal  equations  is  given  by  the  necessary  relation, 

:2pv^-\-  [AV]  =  o, 

which  may  be  proved  in  the  following  manner: 

Let  the  first  equation  in  (14)'  be  multiplied  by  isT,,  the  second 
by  K^,  the  last  by  K,^,,  and  let  the  results  be  added,  giving 

{a,K,  +  fi,K.^  +  .  . .  +  A.A'„0''i  +  K,d, 


+  {a^K^  +   (iqK^  +  .  .  .    4-  \^Kn)Vq  +  Kn'dn'   =  o, 

Now,  as  shown  on  page  61,  the  coefficients  of  z',,  z'j,  .  .  .  v^,  in 
this  equation  are p^v^,  p^''^'^,  .  .  •  pq^^.     Hence 

{p.v^  +/.Z'/  +  . . .  +  A^'/)  +  {KJ,  +  AV2  +  . . .  +  Kn'dn^i  =  o, 

which  may  be  abbreviated  as  is  done  above.  Thus,  if  the 
residuals  be  computed  from  the  observation  equations,  this' 
relation  furnishes  a  check  on  the  numerical  work. 

For  instance,  in   the  example  of  the  preceding  article,   the 
values  of  the  residuals  v  have  been  found.     Then 

2pv^  =  0.3025  -|-  0.0800  +  0.3025  +  0.0225  "i~  0.0225  ~  +  °-73 

Further,  the  values  of  K  and  d  are 

^i  =  +  0.55,  A',  =  -  0.15, 

dj  =   —   1.3,  fl'a  =   +  0.1. 

and  accordingly 

[J^d]  =  +  o.55{-  1.3)  -  0.15(4-  o.i)  =  -  0.73. 

The  necessary  relation  2pv''  -\-  [Kd]  =  o  is  hence  exactly  sat- 
isfied, and  the  numerical  work  may  be  regarded  as  correct. 


§  59-  PROBLEMS.  65 


59.   Problems. 

1.  Six  indirect  observations  upon  two  quantities  furnish  the  fol- 
lowing observation  equations: 

-f  2i  =  +  3.or, 

-\-  IZ2—  —  1.20, 

—  2i  4-  3S2  =  —  4-65, 

-f  23i  —     Zo  —  -\-  6.51, 

2l  +      «2  =   +  2.35. 

2i  —    22  =  i-  3- 70. 

Form  the  two  normal  equations  and  find  the  most  probable  values 
of  s,  and  -2- 

2.  The  bearing  of  a  line  is  taken  five  times  with  a  solar  compass, 
giving  the  values 

A.  12'  E.,     N.  i  E.,     N.  10'  W.,     N.  2'  IV..     N.  2'.qE. 

What  is  the  adjusted  bearing  of  the  line  if  the  weight  of  the  last 
observation  is  five  times  that  of  each  of  the  others? 

3.  Solve  the  following  normal  equations: 

22i  —     22  +  0.52  =  O, 

-  2,  +  420  —      23  —      24  —  0. 26  =  O, 

—  22  +  223  -      24  -|-  0.47  =  O, 

—  22  —      23  +  324  —  35  —    1.08   —  O, 

-  24  +  3-5  +  0-34  =  0. 

4.  A  plane  triangle  has  the  angle  A  measured  ten  times,  B  meas- 
ured five  times,  and  C  measured  once.  The  sum  of  the  three  ob- 
served values  is  found  to  differ  d  seconds  from  180  degrees.  How 
shall  this  d  be  divided  among  the  three  angles? 


66  THE  PRECISION  OF  OBSERVATIONS.  IV. 


CHAPTER    IV. 

THE   PRECISION   OF   OBSERVATIONS. 

60.  In  the  adjustment  of  observations,  it  is  often  necessary 
to  combine  measurements  of  different  degrees  of  precision  ;  and 
for  that  purpose  the  determination  of  their  weights  is  neces- 
sary. When  the  most  probable  or  adjusted  values  have  been 
obtained,  it  is  also  well  to  know  what  degree  of  confidence 
may  be  placed  in  them,  so  that  comparisons  may  be  made  with 
values  obtained  under  other  circumstances.  The  comparison 
of  observations  is  a  very  important  part  of  the  Method  of  Least 
Squares,  since  the  knowledge  of  the  value  and  precision  of 
measurements  is  required  for  their  most  advantageous  use. 
Moreover,  the  study  of  the  precision  of  measurements  is  always 
necessary  to  improve  and  perfect  the  methods  of  observation. 

The  Probable  Error. 

61  The  quantity  usually  selected  to  compare  the  precision 
of  observations  is  the  probable  error,  of  which  the  following  is 
a  definition  : 

In  any  series  of  errors  the  probable  error  has  such  a  value 
that  the  number  of  errors  greater  than  it  is  the  same  as  the 
number  less  than  it.  Or,  it  is  an  even  wager  that  an  error 
taken  at  random  will  be  greater  or  less  that*  the  probable 
error. 


§62.  THE   PROBABLE   ERROR.  6/ 

The  probable  error  is,  then,  the  value  of  x  in  the  probability 
integral  (4)  when  P  =^\,  ox  it  is  the  value  of  x  given  by  the 
equation 


1  2     ^^ 

-  =  -7=  I      e-  '^'^^^d.hx. 

2  VttJo 


By  interpolation  from  Table  I,  Chap.  X,  it  is  found  that 

P  ^=  0.5     when     hx  =  0.4769. 

Hence,  denoting  this  value  of  x  by  r,  the  equation 
(17)  Ar=  0.4769 

gives  the  relation  between  the  measure  of  precision  //  and  the 
probable  error  r,  and  shows  that  h  varies  inversely  as  r. 

62.  To  render  more  definite  the  conception  of  the  measure 
of  precision  h  and  the  probable  error  r,  consider  the  case  of 
two  sets  of  observations  made  with  different  degrees  of  accu- 
racy. Let  the  measure  of  precision  of  the  first  be  Ji„  and  of 
the  second  h^ ;  then,  from  equation  (2),  the  probability  of  errors 
in  the  first  set  will  be  represented  by  a  curve  whose  equation  is 

y  =  h^.dx.ir  —  '^e—^^^^^, 
and  for  the  second  set  by  a  curve 

in  which  dx  is  the  constant  difference  between  two  consecutive 

errors.     Now,  suppose  that  the  second  set  is  twice  as  precise 

as  the  first,  so  that  h,  =  /i,  and  /i^  =  2/1;  then  the  equations 

will  be 

y  =  ahe—''^^^     and    y  =  2  a/i<?— <'*'•'', 

in  which  a  represents  the  constant  ir'^dx.     The  curves  corre- 


68 


THE  PRECISION  OF  OBSERVATIONS. 


IV. 


spending  to  these  equations  are  given  in  Fig.  6;  XB.A^B^X 
being  the  one  for  the  set  of  observations  whose  measure  of 
precision  is  //j  or  //,  and  XB^A^B^X  the  one  for  the  set  whose 
measure  of  precision  is  h^,  or  2k.  These  curves  show  at  a 
glance  the  relative  probabihties  of  corresponding  errors  in  the 


two  sets  :  thus  the  probabihty  of  the  error  o  is  twice  as  much 
in  the  second  as  in  the  first  set ;  the  probabihty  of  the  error 
OP^  is  nearly  the  same  in  each  ;  while  the  probability  of  an 
error  twice  as  large  as  OP^  is  much  smaller  in  the  second  than 
in  the  first  set.  Now,  if  the  lines  P^B^,  P^B^  be  drawn  so  that 
the  areas  P^B^A^B^P^  and  P^B-^A^B-^P^  are  respectively  one- 
half  of  the  total  areas  of  their  corresponding  curves,  the  line 
OP^  will  be  the  probable  error  of  an  observation  in  the  first  set, 
and  OP2  the  probable  error  of  one  in  the  second  set.  Repre- 
senting these  by  the  letters  r,  and  r^,  there  must  be  in  each 
case  the  constant  relation 


hxri  —  0.4769,     h^r^  =  0.4769  ; 


^6$.  THE   PROBABLE   ERROR.  69 

and,  since  h^  is  twice  //„  it  follows  that  r^  must  be  one-half 
of  rj. 

The  probable  error,  then,  serves  to  compare  the  precision  of 
observ^ations  equally  as  well  as  measures  of  precision.  The 
smaller  the  probable  error,  the  more  precise  are  the  measure- 
ments. For  instance,  if  two  sets  of  observations  give  for  the 
length  of  a  line  in  centimeters 

Zi  =  427.32  ±  0.04     and     Z,  =  427.30  ±  0.16, 

in  which  0.04  and  0.16  are  the  respective  probable  errois,  the 
meaning  is,  that  it  is  an  even  wager  that  the  first  is  within  0.04 
of  the  truth,  and  also  an  even  wager  that  the  second  is  within 
0.16  of  the  true  value;  and  the  precision  of  the  first  result  is 
to  be  regarded  four  times  that  of  the  second.  The  probable 
error  thus  serves  as  a  means  of  comparison,  and  also  gives  an 
absolute  idea  of  the  uncertainty  of  the  result. 

63.  In  Art.  43  it  was  shown  that  the  squares  of  measures 
of  precision  are  directly  proportional  to  weights ;  and  in 
Art.  61  it  is  established  that  measures  of  precision  are  in- 
versely proportional  to  probable  errors.  Hence  the  important 
relation  : 

Weights  of  observations  are  inversely  proportional  to  the 
squares  of  their  probable  errors  ;  or,  in  algebraic  language, 

/      r,N  III 

(18)  p^._p^.,p.,.,        .,         : 

Weights  and  probable  errors  are  constantly  employed  in  the 
practical  applications  of  the  Method  of  Least  Squares,  while 
h  is  only  needed  in  theoretic  discussions.  By  means  of  the 
relation  just  established,  the  weights  of  observed  results  of 
different  degrees  of  precision  may  be  found  from  their  computed 


yO  THE   PRECISION  OF  OBSERVATIONS.  IV. 

probable    errors,  and    the    observations    be  thus   prepared  for 
adjustment.     For  instance,  in  the  two  results 

Zi  =  427.32  ±  0.04,     L2  =  427.30  ±  0.16, 

it  is  seen  that  the  weight  of  427.32  is  sixteen  times  that  of 
427.30. 

Probable  Error  of  the  AritJimetical  Mean. 

64.  Let  M^,  M2  .  .  .  Mn  be  n  direct  observations  on  the  same 
quantity.  The  weight  of  each  is  i,  and  the  weight  of  their 
arithmetical  mean  is  ;/.  Let  r  be  the  probable  error  of  a  single 
observation,  and  r^  the  probable  error  of  the  arithmetical  mean. 
The  principle  (18)  of  the  last  article  gives 


I       I 

r  2  ■  r^ 


n:\:: 


from  which 


(19)  r^=  T-; 

V« 

or,  the  probable  error  of  the  arithmetical  mean  is  equal  to  the 
probable  error  of  a  single  observation  divided  by  the  square 
root  of  the  number  of  observations. 

The  probable  error  of  the  mean,  hence,  decreases  as  V^ 
increases.  If  ten  observations  give  a  certain  probable  error 
for  the  mean,  forty  observations  will  be  necessary  in  order  to 
reduce  it  to  one-half  that  value. 

65.  To  find  r,  the  probable  error  of  a  single  observation, 
consider  the  fundamental  law  of  the  probability  of  error  (2),  or 


§65.      PROBABLE   ERROR    OF   THE   ARITHMETICAL    MEAN.      7 1 

By  Art.  12  the  probability  of  the  occurrence  of  the  independent 
errors  ,t-„  x\  .  .  .  x„  is  the  product  of  the  separate  probabilities,  or 

Now,  for  a  given  system  of  errors,  the  most  probable  value  of  h 

is  that   which   has  the   greatest   probability  ;     or  h   must   have 

such  a  value  as  to  render  P'  a  maximum.     Putting  the  first 

.     .       .      dP'  ,  ,       ,     . 

derivative  —--  equal  to  zero,  and  reducing,  gives 
a/i 


71  —  2h-'^x^  =  o,     or     h  = 


-y/^ 


Since,  by  Art.  61,  /ir  equals  the  constant  0.4769, 

0.4769  .  ;'%x^ 

.= -^  =  0.6745  vv 

Here  %x^  is  the  sum  of  the  squares  of  the  true  errors,  which 
are  unknown.  In  a  large  number  of  observations  the  errors 
closely  agree  with  the  residuals,  and  2.r^  may  be  taken  as  equal 
to  ^v^ ;  but,  for  a  limited  number  of  errors,  '^v~  is  less  than  2-r^, 
since,  by  the  principle  of  Least  Squares,  the  first  is  the  mini- 
mum value  of  the  second  ;  so  that 

%x2  =  2z^2  +  u^, 

where  u^  is  a  quantity  as  yet  undetermined.  The  absolute 
value  of  u^  cannot  be  found  ;  but  it  is  known  to  decrease  as  n 
increases,  and  for  a  given  number  of  residuals  to  increase  when 
'^x^  increases  :  as  the  best  approximation,  u^  may  be  taken  as 

equal  to 


It 

1  ncii 

n 

or 

1x^         ^v^ 
n         «  —  I 

"Jl  THE  PRECISION  OF  OBSERVATIONS.  IV. 

and,  inserting  this  in  the  above  value  of  r,  it  becomes 


(20)  r  =  0.6745V/ —^. 

»  «  —  I 


This  is  the  formula  for  the  probable  error  of  a  single  direct 
observation,  or  of  an  observation  of  the  weight  unity.  To  use 
it,  the  residuals  are  to  be  found  by  subtracting  each  measure- 
ment from  the  arithmetical  mean,  and  the  sum  of  their  squares 
then  formed.  When  r  is  known,  the  probable  error  r^  of  the 
arithmetical  mean  is  found  by  the  formula  (19),  or  it  may  be 
written  at  once 


(21)  ro  =  0.6745./ — ?!!! — 

which  is  the  usual  form  for  computation. 


Probable  Envr  of  the  Geiieral  Mean. 

66.  Let  J/,,  ^^-,  .  .  .  Jl/„  be  ;/  direct  observations  having  the 
weights  /„  p2  .  .  .  p„.  The  weight  of  the  general  mean  is 
/i  4-/2  +  •  •  •  -\-  pm  or  2/.  Let  r  be  the  probable  error  of  an 
observation  of  the  weight  unity,  and  r^  the  probable  error 
of  the  mean.  Then,  from  the  fundamental  relation  between 
weights  and  probable  errors, 

r^     r,2 


o 


from  which  the  probable  error  of  the  mean  is 

and,  in  general,  the  probable  error  of  any  observation  is  equal 
to  r  divided  by  the  square  root  of  its  weight.     To  find  r,  an 


§  6/.  PROBABLE   ERROR    OF   THE    GEXERAL    MEAiV.  73 

investigation  like  that  in  the  preceding  article  could  be  em- 
ployed ;  but  it  may  be  well  to  give  one  of  a  different  character. 

67.  Let  h  be  the  measure  of  precision  of  an  observation  of 
the  weight  unity,  and  //,,  //,...  //„  those  of  the  observations 
whose  weights  are/,,/,  .  .  . /„.  By  formula  (7)  the  quantities 
^„  h^  .  .  .  hn  may  be  expressed  in  terms  of  the  weights,  thus  : 

h^^  =  pji-,     h,^  =  pji^  .  .  .  h,c  =  pji-  ; 

and,  in  general,  if  x  be  any  error,  /  the  weight  of  the  corre- 
sponding observation,  and  //  the  measure  of  precision  of  an 
observation  of  the  weight  unity,  the  probability  j  is,  from  (2), 

y  =  hp''iTr~'^-dx.e~  ''-P'^'- . 

Now,  the  quantity  Ipx^y  is  the  same  as  -^^-^,  since  each  term, 

// 

such  as  p^x^,  occurs  n}\_  times  in  //  observations  ;  and,  for  a  con- 
tinuous series  of  errors, 

n  y/TT  J  -«> 

Taking  in  this  hxsjp  =:  t  as  the  unit  variable,  it  may  be 
written 

The  value  of  the  integral  in  this  expression  is  — ,*  and  hence 

%pX^_      I 

n  2/1^ 


*  From  the  footnote  to  Art.  31, 

o  2 


74  THE  PRECISION  OF  OBSERVATIONS.  IV. 

'/^       \o.4769/ 


From  Art.  6i  the  value  of  ~  is  { —  )  :  hence 


=  o.6745V/-i- 


is  the  probable  error  of  an  observation  of  the  weight  unity. 

Now,  2/A-^  is  in  terms  of  the  true  unknown  errors,  and  is 
greater  than  2/t^^     Place,  then, 

2/x=  =  ^pv^  +  «% 

in  which  ii^  is  a  quantity  to  be  determined.  The  probability,  P\ 
of  the  system  of  errors,  is 

P'  =  Ke-'^^^f'^''  =  A'«r~''''(2/>z'=  +  «=)  =  A"<?~^''«\ 

Here  it  is  seen  that  the  law  of  probability  of  ?/^  is  similar  to 
that  of  an  error  x ;  and,  as  in  Art.  31,  it  may  be  shown  that  the 
constant  K'  is  h.Tr~'^du.  The  mean  of  all  the  possible  valu-^s 
of  u^  is,  then, 

/^  r^"       ,,  3 ,  I 

V/tt      "  2h- 


Placing  /  =  t\ s,  this  becomes 

/•oo  Vtj- 

/    e-t^sdt  ~  — P-. 
Differentiating  .his  equation  with  reference  to  s,  and  regarding  t  as  constant 


-/: 


-t^fdsdt  = 


Dividing  this  by  — ds,  and  making  j  =  i,  it  becomes 

e  -t^t^di  —  —  =  one-half  of  the  integral  above, 
o  4 


^68.  LAWS  OF  PROPAGATION  OF  ERROR.  75 

and  this  must  be  taken  as  the  best  attainable  value  of  «'.     But 
it  was  shown  that  —  is  equal  to  -^^.     Hence 

^px~  =  ^pv~  4-  -^—i 
n 

trom  which 


n  n  —  \ 


and  therefore  the  probable  error  r  becomes 


(23)  r=  0.6745  i/-^^. 

V  «  —  I 

The  probable  error  of  the  general  mean  is  now,  from  (22), 


(24)  ''°=°-6745v/7;7l^- 


(;/-  1)2/ 

If  the  observations  be  all  of  the  weight  unity,  2/  becomes  «, 
and  the  formulas  (23)  and  (24)  agree  with  (20)  and  (21).  The 
probable  error  of  any  observation  whose  weight  is  /  is  found 
by  dividing  r  by  the  square  root  of  /. 


Laws  of  Propagation  of  Error. 

68.  Let  z^  and  z^  be  two  independently  measured  quantities 
whose  probable  errors  are  r,  and  r^.  It  is  required  to  find  the 
probable  error  R  of  the  sum  s,  +  Z:,,  or  of  the  difference  z,  —  z^. 
Let  ^  =  ^,  dr  z.,,  and  let  the  errors  arising  in  the  two  cases  be, 

Xi ,  X,",  X,'",  etc.,  for  2,, 
x/,  X2",  X2" ,  etc.,  for  Zj. 


76  THE   PRECISION  OF  OBSERVATIONS.  IV. 

Then  the  corresponding  errors  of  Z  are 

X'  =  xi  ±  x;,     X"  =  xi'  ±  X,",     X'"  =  xl"  ±  x:'\  etc. 

Squaring  each  X,  and  adding  the  results,  gives 

2^2  =  2x1^  ±  2^x^x^  +  %x^^. 

The  products  x^x^  will  be  both  positive  and  negative,  and,  on 
the  average,  "^x^x^,  =:  o  :  hence 

2X^  =  -tx^  +  2^2^ ; 

and,  if  ;/  be  the  number  of  errors, 

2X2  ^  2x,2       2^2^ 
n  n  n 

2r^ 
Now,  by  Art.  65,  it  is  known   that  -^-  varies  with  r^  :  hence, 

11 

for  the  case  in  hand, 

(25)  R^  =  r^  +  r^^; 

from  which  the  probable  error  of  Z  is  known. 

In  like  manner,  if  Z  be  the  sum  or  difference  of  several  inde- 
pendent quantities,  namely,  if 

^  ^^  2 J  It  %2  ni  2-j  jZ  •  •  •  i  ^w/j 
then  the  probable  error  of  Z  is  given  by  the  relation, 

(26)  R^  =  r^  +  ^3^  +  /-j^  +  .  .  .  +  r„,^ 

This    formula   is  very  important    in    the  discussion  of   linear 
measurements. 


§69.  LA^VS   OF  PROPAGATION  OF  FKROK.  J  J 

69.  Secondly,  consider  Z  to  be  a  multiple  of  an  observed 
quantity  z,  so  that  Z  =  Az,  where  ^  is  a  known  number.  Then 
an  error  x  \w  z  produces  an  error  Ax  in  Z,  and 

X  =  Ax,     X^  =  A^x^,    and   2^^  =  A^'^x^. 

Hence,  as  before,  it  is  to  be  concluded  that 

(27)  R^  =  A^r^,  ox  R=  Ar. 

By  combining  the  principle  of  the  last  article  with  that  just 
deduced,  it  is  seen,  if 

Z=  Az,±  Bz^  ±  Cz^  ±  etc., 

and  if  the  probable  errors  of  ^„  z^,  z^  are  r„  r^,  r^,  that  the  prob- 
able error  of  Z  is  given  by 

(28)  R^^  =  A^r^  +  B^r^^  +  C^r^^  +  etc., 

which  is  a  more  general  formula  than  (26). 

It  is  interesting  to  note  that  formula  (19)  can  be  deduced 
from  (26),  and  also  (22)  from  (28).  Thus,  if  z„  z^  .  .  .  z„  are  n 
observed  values  of  the  same  quantity,  the  probable  error  of 
their  sum  is,  by  (26), 


and  by  (27)  the  probable  error  of  ^th  of  this  sum  is 


n  ^n 


which  is  the  probable  error  of  the  arithmetical  mean,  as  in  (19). 


yS  THE  PRECISION  OF  OBSERVATIONS,  IV. 

70.  Next,  consider  Z  to  be  the  product  of  two  independently 
observed  quantities  z,  and  ^„  whose  probable  errors  are  r,  and 
.•-,.  Let  X  be  an  error  in  Z  corresponding  to  the  errors  x^ 
md  x\  in  z^  and  ^^ :  then 

Z  H-  X  =  (zx  +  -^i)  (^2  -^  ^^2)  =  2.^2  +  2i-^2  +  22A*.  +  •^.•^2- 

Here  Z=s^S:,,  and  av^z  vanishes  in  comparison  with  z^^  and 
z^x^ ;  so  that 

^    =   Z1X2,  "r"   22.Xi. 

Squaring  each  error  X,  and  taking  the  sums,  gives 

the  last  term  of  which  vanishes,  since  the  product  x^x^  is  as 
likely  to  be  positive  as  negative :  hence 

and  accordingly,  as  in  Art.  68, 

(29)  R^  =  z,^r^^  +  ViS 

from  which  the  probable  error  of  Z  may  be  computed. 

71.  Lastly,  let  Z  be  any  function  of  the  independently  ob- 
served quantities  z„  z^,  z^  .  .  .,  or  Z  =/{z„  z^,  z^  .  .  .),  and  let  it 
be  required  to  find  the  probable  error  i^  of  Z  from  the  proba- 
ble errors  r„  r^,  r^  .  .  .  of  the  observed  quantities.  Take  .r., 
x^,  x^  as  any  errors  in  ^.,  xr„  z^,  and  X  as  the  corresponding  error 
in  Z:  then 

Z-fX=/[(2, +x,),  (^2  +  ^2),  (zj  +  ^j)---]- 
Now,  if  these  errors  are  so  small,  that  their  second  and  higher 


§72.  ERRORS  FOR   INDEPENDENT  OBSERVATIONS.  79 

powers  may  be  neglected,  the  development  of  the  function  by 
Taylor's  theorem  gives 

^      dZ  dZ  dZ 

dZi  dz^  dz-^ 

Accordingly,  by  the  same  reasoning  as  in  the  previous  articles, 


<->-  =  (f)-Kf)'-( 


dZ\2 


which  is  a  general  formula  appplicable  to  all  functions. 

The  laws  of  propagation  of  error,  given  by  formulas  (25)  to 
(30),  are  very  important  in  forming  proper  rules  for  taking 
observations,  as  well  as  in  discussing  and  comparing  results. 

The  law  R  =  V^,^  +  ^2%  which  gives  the  probable  error  of  Z 
when  Z  =^  c^  -\-  z^,  or  when  Z  ^=  z^  —  z^,  has  been  likened  by 
Jordan  to  the  celebrated  geometrical  theorem  of  Pythagoras. 

Probable  Errors  for  Independent  Observations. 

72.  In  Arts.  46-50  are  given  methods  of  finding  the  most 
probable  values  of  independent  quantities  which  are  indirectly 
observed.  To  determine  the  probable  errors  of  any  adjusted 
value,  z,  let  p^  denote  its  weight,  and  r^  its  probable  error. 
Then,  if  rbe  the  probable  error  of  an  observation  whose  weight 
is  unity,  the  relation  (18)  gives 


pt  :  1  :  : 
from  which 

(31)  ^z  = 


I       I 


Hence,  in  order  to  find  the  probable  errors  of  zi,  z^  .  .  .  z  ,  it  is 


80  THE   PRECISION  OF  OBSERVATIONS.  IV, 

necessary  to  find  r  and  their  weights.  And,  in  general,  the 
probable  error  of  any  observation  is  equal  to  r  divided  by 
the  square  root  of  its  weight. 

73.  To  find  the  probable  error  of  an  observation  whose 
weight  is  unity,  the  following  reasoning  may  be  employed  : 

Suppose  that  the  normal  equations  (13)  have  been  solved,  and 
the  most  probable  values  ^,,  z^  .  .  .  ^q  deduced.  Let  the  corre- 
sponding true  values  be  represented  by  ^,  -|-  ^-i.  -2  +  ^-^2  •  •  • 
Zq  +  8^^,  in  which  S^i,  Iz^  .  .  .  hz^  are  small  unknown  correc- 
tions. Now  if,  in  the  observation  equations  (12),  the  most 
probable  values  be  substituted,  they  will  not  reduce  to  zero, 
but  leave  small  residuals  z\,  v^  .  .  .  v,^ ;  thus  : 

fl.z,  +  b^Z:,  +  .  .  .  +  /,c^  +  m,  =  v^    with  weight/,, 
a^Zi  +  b^Zz  4-  .  .  .  +  A-?  +  '''^  =  ^'2,  with  weight /^j 


^«2i  +  bnZ:,  +  .  .  .  +  l,fiq  +  m„  =  v„,  with  Weight  />, 


fi} 


while,  if  the  corresponding  true  values  be  substituted,  they  will 
give  the  true  errors  ;  thus  : 

a,  (s.  +  &,)  +  b,  (S2  +  8S2)  +  .  .  .  +  w,  =  X,, 
«2  (2i  +  Ssj)  +  b.  (Sj  +  822)  +  .  .  .  +  m^  =  X2, 


an{z,  +  8s,)    +  b„{z.  +  8^2)   +  .   .  .  +  W3   =  Xn. 

Subtracting  each  one  of  the  former  equations  from  the  latter 
gives  the  following  residual  equations  : 

v^  -f  «,82,  -f  ^,8^2  4-  .  .  .  +  /,8s^  =  x„ 
V2  +  a2^Zi  +  ^,802  +  .  .  .  +  I^^Zg  =  X2, 


v„  -\-  a„lz^  +  b„lz2  +  .  .  .  -h  l„lZq  =  Xn. 

Now,  the  principle  of  Least  Squares  (6)  requires  that  ^px^  shall 
be  made  a  minimum  to  give  the  most  probable  values  of  z^ 


^73-  E JURORS  FOR   INDEPENDENT  OBSERVATIONS.  Si 

z^  .  .  .  Zg;  and,  by  the  solution  of  the  normal  equations,  its  mini- 
mum value  is  the  sum  2/I^^  From  the  residual  equations  a 
relation  connecting  the  two  sums  tpv^  and  l.px^  may  be  found 
by  squaring  both  members  of  each  of  those  equations,  multi- 
plying each  by  its  corresponding  weight,  and  then  adding  the 
products.  Without  actually  performing  these  operations,  it  is 
evident,  that  if  the  squares  and  products  of  lz„  Iz^  .  .  .  hZg  be 
neglected  as  small  in  comparison  with  the  first  powers,  the 
result  will  be  of  the  form 

2/z/^  -f  kM,  +  K^z^  +  .  .  .  +  kglzq  =  ipx^, 

in  which  k^,  /%  .  .  .  k^  are  co-efficients  of  the  unknown  correc- 
tions, and  dependent  only  upon  the  known  co-efficients  and 
weights.  If  the  number  of  unknown  quantities  is  q,  there  will 
be  q  of  these  terms.     Placing 

it  becomes 

Now,  the  probability  of  the  occurrence  of  the  error  x„  whose 
measure  of  precision  is  Ji„  and  whose  weight  is  /,,  is,  by  (2) 

and  (7), 

y,  —  hpih.dx.Tr—he-''^P^'^^'^, 

in  which  //  is  the  measure  of  precision  of  an  observation  of 
the  weight  i.  And  hence,  by  exactly  the  same  reasoning  as 
in  Art.  6j,  it  may  be  shown,  that,  when  n  is  a  large  number, 

%px-  =  \ 

Further  :  if  there  be  but  one  unknown  quantity,  there  is  but 

one  li^t  whose  value,  as  shown  in  Art.  Gj,  is  — .     And,  since 

2h^ 


82  THE  PRECISION  OF  OBSERVATIONS.  IV. 

this  is  true  whichever  unknown  quantity  be  considered,  the 
value  of  each  ii^  must  be  — ;  and,  as  there  are  a  of  these 
values,  the  above  result  becomes 


from  which 


In 


~q 


^pv^ 


Therefore,  from  the  constant  relation  (17)  between  h  and  r, 
the  probable  error  of  an  observation  of  the  weight  unity  is 


(32)  r=  0.6745 


74.   The  probable  errors  of  the  values  ^„  z^  .  .  .  Zg 
be  found  from  (31)  as  soon  as  their  weights  are  known.     These 
will  now  be  determined. 

The  observations  J/„  M^  .  .  .  JMn  furnish  the  observation 
equations  (12)  and  the  normal  equations  {13).  The  solution  of 
the  latter  gives  the  values  of  z^,  z^  .  .  .  Zg  in  terms  of  J/„ 
M2  .  .  .  M„,  and  co-efficients  independent  of  those  quantities. 
Suppose  the  general  solution  to  give 

2,  =  o-,J/.  +  o-,yI/,  +  0-3J/3  +  •  . .  +  o-„J/«, 

2,    =    tJI,   +   tJI,    +   T3J/3    +  .    .   .   +  TnM„, 


in  which  the  co-efificients  a,  t  .  .  .  ^  depend  only  upon  the  con- 
stants a,  b  ...  I  and  the  weights  /,,  p2  ■  ■  •  pn-  Then,  if  R^^  is 
the  probable  error  of  .::,,  and  ;-,,  }\  .  .  .  r„  are  the  probable  errors 
of  J/.,  M^  .  .  .  M„,  the  formula  (28)  gives 


74- 


es:kors  of  independent  observations. 


83 


Now,  since  the  squares  of  probable  errors  are  inversely  pro- 
portional to  weights,  this  becomes 

A:~A     '^  p.     ^'"^  Pn    ~IP   S 

from  which  the  value  of  p^^  is 

Pn  = 


<J(T~\' 

-T  A 


In  like  manner  it  is  easy  to  show  that  the  weight  of  s^  is  the 

LPS 


,  and  that  the  weight  of  s^  is  the  reciprocal 


of 


reciprocal  of 

"CC" 

-/■ 

Owing,  however,  to   the    labor   of    finding  the   co-efficients 

o",  r  .  .  .  C,  it   is   better   to   deduce    these  expressions  under  a 

different  form.     Suppose  the  normal  equations  to  be  solved, 

giving 

z^  =  a,[/>aAf]  +  a2[pdAf]  +  .  .  .  +  a^[/>/Af], 

z,  =  §ipaM-\  +  lilpbM\  +  .  .  .  +  ftlplM\ 

z,  =  X,[/>aA/]  +  A,[/^J/]  +  .  .  .  +A,[//J/], 

in  which    a,  ^  .   .    .A    are    co-efficients   independent    of    M„ 
M2  .  .  .  M„.    Then  the  respective  weights  of  z,,  z^  .  .  .  z^  will  be 

— ,  —  ...  --.     To   show  this,  it  will   be  sufficient  to  consider 
a,     fi^  A, 

the  quantity  z^,  and  to  prove  that      —      =  /J^.     By  comparing 
the  above  two  expressions  for  z^,  it  is  seen  that 
r,  =  ft,p,a^  +  §p,b,  +  .  .  .  +  /?^/,/., 


^«   =  /5'i  />«««  +  §2i>nbn  +   •   •   .   +   ftlpnln- 


84  THE   PRECISION   OF   OBSERVATIONS.  IV. 

Squaring  eadh  of  these  equations,  dividing  each  by  its/,  and 
adding  the  results,  gives 


rr 

l7j 


=  A(/?.[/^«]  +  ftlpab-\  +  .  .  .  +  A[/^/]) 

+  §.{ftlpab\  +  ii.lpbb\  +  .  .  .  +  /5,[#/])  4-  .  . . 

+  ft.^ftlpan  +  A[/^/]    +  .  .  .  +  ^,[///]). 

Now,  if  the  normal  equations  (13)  are  solved  by  the  method  of 
undetermined  multipliers,  the  first  is  to  be  multiplied  by  a 
number  ji^ ,  the  second  by  ft^,  the  ^"'  by  /^^,  and  the  products 
added.  Then,  if  upon  these  multipliers  the  following  condi- 
tions be  imposed, 

§,{paa\  +  ftlpab\  +  .  .  .  +  ftlpal\  =  o, 
P,[pab]  +  /i,[pbb]   +...  +  MpM]  =  I, 


/3,[pa/]   +  ^Ipb/]  +...  +  /?,[///]   =  o, 

all  the  terms  except  those  involving  /J^  will  reduce  to  zero,  and 
the  value  of  /?,  will  be  the  same  as  above  expressed.  Accord- 
ingly \  ~T  =  y^2>  and  the  weight  of  -s^j  is  -7- ;  which  was  to  be 
proved. 

75.  The  following  is  hence  a  method  of  finding  the  weights 
of  the  values  of  the  unknown  quantities.  Preserve  the  abso- 
lute terms  of  the  normal  equations  in  literal  form  during  the 
solution.  Then  the  weight  of  any  value,  as  xTj,  is  equal  to 
the  reciprocal  of  the  co-efificient  of  the  absolute  term  which 
belonged  to  the  normal  equation  for  ^-3. 

For  example,  take  the  normal  equations 

—  ^i  ~r  3^2  —    ^3  ^^  ■"2> 


^y6.  EKKORS   OF  INDEPENDENT  OBSERVATIONS.  85 

The  solution  of  these  by  any  method  gives 

2i  =  f^,  +  %A^  +  1^3, 

Z3  =  \A,  +  i^,  +    ^3,  . 

and  hence  the  weight  of  z,  is  |,  the  weight  of  z^  is  f,  and  the 
weight  of  ^3  is  I.  It  is  evident,  if  it  be  only  desired  to  find 
the  weight  of  ^„  that  A^  and  A^  need  not  be  retained  in  the 
computation,  but  may  be  made  zero.  So,  in  finding  the  weight 
of  z^^  only  A^  need  be  retained  in  the  work, 

76.  As  an  illustration  of  the  preceding  principles,  let  there 
be  three  observation  equations  of  weight  unity, 

2,  =  0,     ^2  =  0,     Si  —  22  =  +  0.51. 

The  normal  equations  are 

2^1   —  2:,   =   +  0.51,       —  2,  +  23'2  =    —  0.51. 

Writing  A^  and  A^  for  the  absolute  terms  the  solution  of  these 
equations  gives 

21  12 

2,  =  -A^  +  -A^,        ^2  =  -A^  +  -^„ 

3  3  3  3 

from  which  the  adjusted  probable  values  are  z^  =  -}-0. 17  and 
^2  =  —  0.17,  while  the  weight  of  each  of  these  values  is  seen 
to  be  i^.  The  sum  of  the  squares  of  the  residuals  is  ^f- = 
0.0867,  ^"d  from  (32)  the  probable  error  of  an  observation  of 
weight  unity  is  ±  0.20.  This  divided  by  V1.5  gives  ±  0.16  as 
the  probable  error  of  the  adjusted  values  of  s^  and  z^.  The 
adjusted  value  of  the  third  observation  is  2,-22=  +0-34)  and 
by  (25)  the  probable  error  of  this  value  is  ±0.23.  It  is  seen 
that  the  corrections  to  the  three  observed  values  are  here 
numerically  equal. 


86  THE  PRECISION  OF  OBSERVATIONS.  IV. 


Probable  Errors  for  Conditiotied  Observations. 

77.  When  conditioned  observations  are  adjusted  by  the  gen- 
eral method  of  Art.  57,  where  the  q  unknown  quantities  in 
the  n  observation  equations  are  reduced  to  q  —n'  independent 
quantities  by  means  of  the  n'  conditional  equations,  the  proba- 
ble error  of  an  observation  of  the  weight  unity  is  evidently 
given  by  the  formula  (32),  if  q  be  replaced  by  q  —  n',  or 


(33)  '■=°■''^^s\/-,J^'+^■■ 


and  the  probable  errors  of  observations  or  values  whose  weights 
are/,,/2,  etc.,  are,  by  (31), 

r  r 

r,  =  ^=,     r^  =  -=,  etc. 

The   weights   of   ^„  z^  .  .  .  Zq  are  to   be   found    exactly  as   in 
Art.  75. 

For  the  case  of  direct  observations  on  several  quantities 
adjusted  by  the  method  of  Art.  58,  the  number  of  observation 
equations  is  the  same  as  that  of  the  unknown  quantities,  or 
n  =  q\  and,  if  «'  be  the  number  of  conditional  equations,  the 
probable  error  of  an  observation  of  the  weight  unity  is 


(34)  ^=o.6745y-^f-, 


from  which  the  probable  error  of  any  observation  of  given 
weight  can  at  once  be  deduced.  In  this  case  the  residuals  v 
are  merely  the  differences  between  the  observed  and  the 
adjusted  values. 


§  78.  PROBLEMS.  87 


78.   Problems. 

1.  There  are  two  series  of  observations  of  an  angle,  each  taken  to 
hundredths  of  a  second.  The  probable  error  of  a  single  observation  in 
the  first  series  is  o".65,  and  in  the  second  i".45.  Compute  the  proba- 
bilities of  the  error  o".oo  and  of  the  error  2".oo  in  the  two  cases. 

2.  It  is  required  to  determine  the  value  of  an  angle  with  a  proba- 
ble error  of  o".25.  Twenty  measurements  give  a  mean  whose  probable 
error  is  o".38.     How  many  additional  measurements  are  necessary? 

3.  Find  the  probable  error  of  the  mean  of  two  observations  which 
differ  by  the  amount  a. 

4.  Let  z„  ^2,  and  z^  be  independently  observed  quantities  whose 
probable  errors  are  r„  r^,  and  r^  If  Z=  s,^  -j-z^^  _|_  ^^z  fl^^  the  proba- 
ble error  of  Z. 

5.  Let  r  be  the  probable  error  in  log  a.  What  is  the  probable  error  in 
the  number  a  ? 

6.  Given  the  following  observation  equations  :  — 

2,  =  4.5,     with  weight  10, 

Z2  =  1.6,      with  weight    5, 

2,  —  32  =  2.7,     with  weight    3. 

What  are  the  most  probable  values  of  2.  and  z^  with  their  probable 
errors  ? 

7.  Given  the  observation  equations  (all  of  equal  weight) 

30,  +  322  —    Zi=  14, 
42,  -F   22  -t-  423  =  21, 

—  52,  +  222  +  333  =    5, 

to  find  the  best  values  of  z^,  z^,  and  z^,  with  their  probable  errors. 


S8         DIRECT  OBSERVATIONS   ON  A   SINGLE    QUANTITY.         V. 


CHAPTER  V. 

DIRECT   OBSERVATIONS   ON   A    SINGLE   QUANTITY. 

79.  In  the  preceding  pages  the  fundamental  methods  and 
formulas  for  the  adjustment  and  comparison  of  observations 
have  been  deduced.  In  this  and  the  three  following  chapters 
the  application  of  these  methods  to  practical  examples  will  be 
presented.  The  most  common  case  of  observation  is  that  of 
direct  measurements  on  a  single  quantity,  and  this  will  form 
the  subject  of  the  present  chapter. 


Obseivations  of  Equal  Weight. 

80.  When  a  quantity  is  measured  several  times  with  equal 
care,  so  that  there  is  no  reason  for  preferring  one  observation 
to  another,  the  observations  are  of  equal  weight.  From  re- 
mote antiquity  the  arithmetical  mean  of  the  measurements 
has  always  been  regarded  as  the  best  or  most  probable  value 
of  the  quantity  sought ;  and,  as  shown  in  Art.  44,  this  is  con- 
firmed by  the  fundamental  principle  of  the  Method  of  Least 
Squares. 

Let  c  be  the  most  probable  value  of  the  measured  quantity, 
n  the  number  of  observations,  and  M  any  observation.  Let  r 
be  the  probable  error  of  a  single  observation,  and  r^  the  proba- 


§8 1.  OBSERVATIONS   OF  EQUAL    UEIGIIT.  89 

ble  error  of  the  adjusted  value  z.     Let  also  v  be  any  residual 
obtained  by  subtracting  M  from  z. 

The  most  probable  value  of  the  quantity  is  the  arithmetical 
mean,  expressed,  as  in  Art.  44,  by  formula  (8), 

z  =  . 

n 

The  probable  error  of  a  single  observation,  as  shown  in  Art. 
65,  is,  by  formula  (20), 


r  =  0.6745 


Lastly,  as  shown  in  Art.  64,  the  probable  error  of  the  mean  is, 

h.y  (19). 

r 

\jn 

Formula  (8)  indicates  the  method  of  adjustment,  while  (20) 
pnd  (19)  determine  the  precision  of  observation  and  of  the 
mean.  After  finding  z,  each  observation  is  subtracted  from  it, 
giving  ;/  values  of  v.  The  squares  of  these  are  taken,  and 
their  sum  is  ^v^ ;  then  r  is  computed,  and  lastly,  r^.  If  desired, 
To  can  be  also  found  directly  from  formula  (21), 


/-o  =  0.6745 


/      %v^ 
S  n{n  -  1) 


which  is  the  same  as  (19). 


81.  As  an  example,  consider  the  following  twenty-four  meas- 
nrem.ents  of  an  angle  of  the  primary  triangulation  of  the 
United-States  Coast-Survey,  made  at  the  station  Pocasset  in 
Massachusetts,  and  recorded  in  the  Report  for  1854: 


QC        DIRECT  OBSERVATIONS  ON  A   SINGLE   QUANTITY.        V. 


Observations. 

V. 

^'^ 

Il6°43'44".45 

5-19 

26.94 

50-55 

—  0.91 

-83 

50-95 

-I-3I 

1.72 

48.90 

0.74 

-55 

49.20 

0.44 

.19 

48.85 

0.79 

•63 

47.40 

2.24 

5.02 

47-75 

1.89 

3-57 

51-05 

-I.4I 

2.00 

47-85 

1.79 

3.20 

50.60 

—  0.96 

.92 

48-45 

1. 19 

1.42 

51-75 

—  2.11 

4  45 

49.00 

0.64 

.41 

52-35 

-2.71 

7-34 

51-30 

-1.66 

2-75 

51-05 

— 1.41 

2:00 

51-70 

—  2.06 

4.24 

49-05 

0.59 

-35 

50-55 

—  0.91 

-S3 

49-25 

039 

•15 

46.75 

2.89 

8.35 

49-25 

0.39 

-15 

53.40 

-3.76 

14.14 

Z  = 

ii6°43'49".64 

2?'2    =     92.15 

The  most  probable  value  of  the  angle  is  found  by  adding  the 
observations,  and  dividing  the  sum  by  twenty-four.  1  his  is 
1 16°  43' 49". 64.     Subtracting  from  this  the  first  reading  gives 


82.  OBSERVATIONS  OF  EQUAL    WEIGHT. 


91 


5.19  for  the  first  residual,  which  is  placed  in  the  column  headed 
V.  The  square  of  this  is  26.94,  which  is  placed  in  the  column 
headed  ^'^  The  sum  of  all  these  squares  is  92.15.  Then  from 
(20)  the  probable  error  of  a  single  observation  is 


0.6745  v/^;-5=. ".35; 


^3 

and  the  probable  error  of  the  mean  is,  from  (19), 

ro  =  -^  =  o".28 : 

hence  the  final  value  may  be  written  116°  43'  49'''.64  d=  o".28. 

The  precision  of  the  mean  of  these  twenty-four  observations 
is  such  that  o".28  is  to  be  regarded  as  the  error  to  which  it  is 
liable  ;  that  is,  it  is  an  even  wager  that  the  mean  differs  from 
the  true  value  of  the  angle  by  less  than  o".28,  and  of  course 
also  an  even  wager  that  it  differs  by  more  than  o".28.  The  pre- 
cision of  a  single  observation  is  such  that  i".35  is  the  error  to 
which  it  is  liable  ;  that  is,  half  the  errors  should  be  less,  and 
half  greater,  than  i".35  in  a  large  number  of  observations.  It 
will  be  noticed  that  twelve  of  the  above  residuals  are  less,  and 
twelve  greater,  than  i".35. 

In  Art.  27  it  was  shown  that  the  algebraic  sum  of  the  residu- 
als must  always  equal  zero.  This  principle  may  be  used  to 
furnish  a  check  on  the  accuracy  of  the  numerical  work. 

82.  The  tables  in  Chap.  X  will  be  found  useful  in  abbreviat- 
ing computations.  By  the  help  of  Table  VI  the  squares  of 
the  residuals  can  be  readily  found.  By  Table  III  the  compu- 
tation of  r  and  ;-„  can  be  much  abridged ;  for  instance,  in  the 
case  of  the  last  article,  n  =:  24,  and 


r  =  0.1406  i/92.15  =  i".35, 
ro  =  0.0287  1/92.15  =  o  -28. 


92         DIRECT  OBSERVATIONS  ON  A   SINGLE   QUANTITY.        V. 

The  table  of  four-figure  logarithms  will  also  prove  useful  in 
extracting  roots  and  performing  multiplications. 

When  the  tables  are  used,  it  will  be  found  more  convenient  to 
compute  To  from  (21)  than  from  (19).  Formula  (19),  however, 
is  very  important  in  indicating  that  the  probable  error  of  the 
mean  decreases,  and  hence  that  its  precision  increases,  with 
the  square  root  of  the  number  of  observations. 

It  should  be  borne  in  mind,  that  the  method  of  the  arithmeti- 
cal mean  only  applies  to  equally  good  observations  on  a  single 
quantity,  and  that  it  cannot  be  used  for  the  adjustment  of  ob- 
servations on  several  related  quantities.  For  instance,  let  an 
angle  be  measured,  and  found  to  be  6oi  degrees,  and  again  let 
it  be  measured  in  two  parts,  one  being  found  to  be  40  degrees, 
and  the  other  20  degrees.  The  proper  adjusted  value  of  the 
angle  is  not,  as  might  at  first  be  supposed,  the  mean  of  60^  and 
60,  which  is  6o\  degrees,  but,  as  will  be  seen  in  the  next  chap- 
ter, it  is  60^  degrees,  —  a  result  which  requires  the  correction  of 
each  observation  by  the  same  amount. 

Shorter  Formulas  for  Probable  Error. 

83.  The  method  of  computing  probable  errors  by  formula  (20) 
is  that  considered  the  best  by  all  writers.  Nevertheless,  on 
account  of  the  labor  of  forming  the  squares  of  the  residuals, 
a  simpler  and  less  accurate  formula  is  often  employed,  in  which 
only  the  residuals  themselves  are  used.  To  deduce  it,  let  71  be 
the  number  of  observations,  and  2z'  the  sum  of  the  residuals, 
all  taken  with  the  positive  sign,  and  2.t'  the  sum  of  all  the  errors 

taken  positively.     Then  —  is  the  mean  of  the  errors ;  and,  by 
the  same  reasoning  as  in  Art.  Gj,  this  mean  is 


%x  2/1  /"»  I 


n 


hsfrt 


§  84-  SHORTER   FORMULAS  FOR   PROBABLE   ERROR.  93 

Now  since,  by  Art.  61,  the  product  //r  is  equal  to  the  constant 
0.4769,  the  value  of  r  in  terms  of  2.f  is 

r  =  0.8453  — 
n 

The  sum  of  the  errors  '%x  is  in  general  different  from  the 
sum  of  the  residuals  ^v.  Both  in  Art.  65  and  Art.  6"]  it  was 
shown  that 

—  > 

n  11  —  \ 

and  it  may  hence  be  concluded,  that,  on  the  average,  x^  is  greater 
than  v^  in  the  ratio  of  ;^  to  ;/  —  i,  and  that,  on  the  average,  x  is 

greater  than  v  in  the  ratio  of  sju  to  ^n  —  i,  or  that 

1,x  2z' 


V^;/         \jn  —  I 

Accordingly  the  above  value  of  r  becomes 

0.84532?; 


(35) 


^n{n  —  i) 


which  gives  the  probable  error  of  a  single  observation.      By 
substituting  this  in  (19),  the  value  of  r^  becomes 

(36)  r„=°:MjlS, 

n^n  —  I 

which  is  the  probable  error  of  the  arithmetical  mean. 

84.  Formulas  (35)  and  l^iG)  will  be  found  much  easier  to  use 
than  (20)  and  (21).  In  Table  IV  the  co-efficients  of  ^v  are 
tabulated  for  values  of  ;/  from  2  to  100,  and  by  its  use  the 
computations  are  much  abridged. 


94 


DIRECT  OBSERVATIONS   ON  A   SINGLE   QUANTITY.        V. 


As  an  example,  consider  the  following  eight  measurements 
of  a  line  made  with  a  tape  twenty  meters  long,  graduated  to 
centimeters  : 


Observations. 

V. 

188.97 

0.095 

.88 

.005 

.91 

•035 

•99 

•115 

.83 

•045 

.80 

.075 

.81 

.065 

.81 

.065 

188.875 

0.500 

Here  the  arithmetical  mean,  or  most  probable  value  of  the  line, 
is  found  to  be  188.875  meters.  The  difference  between  this 
and  the  single  observations  gives  the  residuals  v,  whose  sum 
Iro  =  0.5.     Then,  by  the  use  of  Table  IV,  for  n  =  8, 

r  =  0.1130  X  0.5  =  0.0565, 
fo  =  0.0399  ^  0-5  =  0.0200. 

By  the  more  accurate  formulas  (20)  and  (21)  these  values  are 

r=  0.051     and     fo  =  0.018  meters. 

With  a  larger  number  of  observations,  a  closer  agreement 
between  the  probable  errors  found  by  the  two  methods  might 
be  expected. 

85.   The    probable    error  r  of   a    single    observation    should 
always  be  computed,  since  it  furnishes  the  means  of  comparing 


§86.  OBSERVATIONS   OF   UNEQUAL    WEIGHT.  95 

the  accuracy  of  work  done  with  different  instruments,  or  by 
different  observers.  Under  similar  conditions,  r  should  be  prac- 
tically a  constant  for  a  given  class  of  measurements  ;  while  for 
different  classes  the  different  values  of  r  indicate  the  relative 
precision  of  the  methods.  For  instance,  suppose  the  same 
observer  to  measure  the  same  angle  with  two  different  transits, 
and  to  find  the  probable  error  of  a  single  observation  with  the 
first  to  be  4",  and  with  the  second  6".  The  relative  precision 
of  the  instruments  is,  then,  inversely  as  these  probable  errors, 
or  as  3  to  2  ;  and  the  weights  of  a  single  observation  in  the  two 
cases  are  as  3^  to  2^  or  as  2|  to  i  ;  so  that  one  measurement 
made  with  the  first  instrument  is  worth  2\  made  with  the 
second.  These  results,  in  order  to  be  satisfactory,  must  be 
deduced  from  a  large  number  of  observations  ;  since  the  formu- 
las for  probable  error  suppose  that  enough  observations  are 
made  to  exhibit  the  several  residuals  according  to  the  law  of 
probability  of  error  as  given  by  equations  (i)  and  (2). 

Observations  of  Unequal   Weight.  \ 

\ 
86.   When  the  observations  on  a  single  quantity  have  differ-        i 

ent  weights,  the  most  probable  value  of  the  quantity  is  to  be         ^ 

found  by  the  use  of  the  general  arithmetical  mean ;  namely,  by 

multiplying  each  observation  by  its  weight,  and  dividing  the         ! 

sum  of  the  products  by  the  sura  of  the  weights.     Or  if  .?  be 

that  most  probable  value,  M  any  observation,  and  /  its  weight, 

then,  as  shown  in  Art.  45,  formula  (9)  gives 

2/*  : 

The  probable  error  of  an  observation  of  the  weight  unity,  as         I 
shown  by  formula  (24),  Art.  67,  is  I 


t=  0.6745  v,    ^' 

T    («  —   I 


(«-l) 


96        DIRECT  OBSERVATIONS  ON  A    SINGLE   QUANTITY.        V. 

in  which  n  denotes  the  number  of  observations,  and  v  any 
residual  obtained  by  subtracting  M  from  ;:.  Lastly  the  proba- 
ble error  of  z,  as  shown  in  Art.  66,  is  found  by  (22), 


r^  = 


v/2/ 

Formula  (9)  indicates  the  method  of  adjustment.  Having 
found  the  most  probable  value  z,  each  observation  is  subtracted 
from  it,  giving  «  residuals  v.  These  are  squared,  and  each  v^ 
multiplied  by  the  corresponding  weight  /.  The  sum  of  these 
products  is  2/■^'^  Then  formula  (24)  gives  the  probable  error 
of  an  observation  of  the  weight  unity.  Lastly,  formula  (22) 
gives  the  probable  error  of  .::.  And  in  general  the  probable 
error  of  an  observation  of  given  weight  may  be  found  by  divid- 
ing r  by  the  square  root  of  that  weight. 

87.  As  an  example  let  the  observations  in  the  second  column 
of  the  following  table  be  the  results  of  the  repetition  of  an  angle 
at  different  times,  18". 26  arising  from  five  repetitions,  16", 30 
from  four,  and  so  on,  the  weights  of  the  observations  being 
taken  the  same  as  the  number  of  repetitions.  Then  the  general 
mean  £•  has  the  weight  21,  the  sum  of  the  several  weights  or 


p- 

M. 

V. 

vK 

/>v^. 

5 
4 

I 

4 
3 

4 

87°  51'  i8".26 
16.30 
21.06 

17-95 
16.20 

20.85 

—  0. 10 
-\-  1.86 

—  2.90 

+  0.21 

+  1.96 

—  2.69 

O.OIO 

3.460 
8.410 
0.044 

3.842 
7.236 

0.05 

13.84 

8.41 
0.18 

11-53 
28.94 

^/=  21 

2  =  87°  51'  i8".i6 

^pv''  =  62.95 

§88.  OBSERVATIONS    OF    UNEQUAL    WEIGHT.  97 

the  number  of  single  measures.  Subtracting  each  M  from  z 
gives  the  residuals  in  the  column  v\  next  from  Table  VI  the 
numbers  in  the  column  v^  are  found,  and  multiplying  each  of 
these  by  the  corresponding  weight  produces  the  quantities/?/, 
whose  sum  is  62.95.     Then,  since  )i  is  6,  formula  (23)  gives 


r  =  0.6745 


or,  by  the  help  of  Table  III, 


r  =  0.3016  r  62.95  ^^  2    .39. 

This   is  the  probable  error  of   an    observation    of   the   weight 
unity.     From  (22)  the  probable  error  of  the  general  mean  is. 


r  21 


and  the  probable  error  of  any  given  observation  is  found  by 
dividing  2^.39  by  the  square  root  of  its  weight. 

88.  The  important  relation  (i8)  of  Art.  63,  that  the  weights 
of  observations  are  inversely  as  the  squares  of  their  probable 
errors,  furnishes,  as  already  indicated  in  Art.  85,  a  ready  means 
of  determining  weights,  if  the  probable  errors  can  be  obtained 
with  sufficient  precision.  When  the  weights  are  known,  the 
observations  can  be  combined  by  (9),  and  the  most  probable 
value  determined. 

As  an  example,  consider  the  two  following  series  of  meas- 
urements of  an  angle  ;  the  first  taken  with  a  transit  reading 
to  twenty  seconds,  and  the  second  with  a  transit  reading  to 
minutes.     The  angle  was  observed  in  each  case  ten  times;  the 


98        DIRECT  OBSERVATIONS  ON  A   SINGLE   QUANTITY.        V. 

circle  being  used  in  eleven  different  positions  to  eliminate  errors 
of  graduation,  while  each  time  the  two  verniers  were  read  to 
eliminate  errors  of  eccentricity. 


With  First  Transit. 

With  Second  Transit. 

M. 

- 

v. 

v". 

M. 

TO. 

u^ 

34    55  35 
35 

20 
05 

75 
40 

10 

30 
50 
30 

2 
2 

13 

28 

42 
7 

13 
3 

17 
3 

4 

4 

169 

784 

1764 

49 
169 

9 

289 

9 

34°  56'  15" 
55  30 

54  30 

55  15 

56  00 

55  45 
55  30 

55  30 

56  CO 

55  45 

39 
6 

66 

21 

24 

9 
6 

6 

24 

9 

1521 

36 

4356 

441 

576 
81 

36 

36 

576 

81 

34° 55' 33" 

3250 

34°  55'  36" 

7740 

By  the  method  of  Art.  80  it  is  easy  to  find 


For  first  transit  . 
For  second  transit 


34°  55'  Zl"  ±  4".i 
34    55    36     ±6    .3 


Hence  by  (18)  the  weights  of  these  means  are  in  the  ratio 


—  :  — ,     or  as  12  to  5  nearly. 

41^    63^  ^ 

The  final  adjusted  value  of  the  angle  is,  then, 

„      ,,    31  X   12  +  36  X  5  o      ,      ,, 

■^  =  34    55    + ;y^ =  34    55    33   •9» 


§  89.  PROBLEMS.  99 

and  by  (18)  the  probable  error  of  that  value  is 


ro  =  4.1 


n/:-; = 3".4. 


As  the  probable  errors  of  a  single  observation  in  the  two  cases 
are  13"  and  20",  the  corresponding  weights  are  as  400  to  169; 
so  that  one  observation  with  the  first  instrument  is  worth  about 
2\  with  the  second. 

When  observations  upon  the  same  quantity  are  known  to  be 
of  different  precision,  and  there  is  no  way  of  finding  the  proba- 
ble errors,  as  in  the  example  just  discussed,  weights  should  be 
assigned  corresponding  to  the  confidence  that  is  placed  in 
them,  and  then  the  general  mean  can  be  deduced.  Of  course, 
the  assignment  of  weights  in  such  cases  is  a  matter  requiring 
experience  and  judgment. 

Problems. 

89.  The  solution  of  the  following  problems  will  serve  to 
exemplify  the  preceding  principles. 

1.  The  latitude  of  station  Bully  Spring,  on  the  United  States  northern 
boundary,  was  found  by  sixty-four  observations  to  be  49°  01'  09".  11 
±  o".05i.     What  was  the  probable  error  of  a  single  observation? 

2.  A  line  is  measured  five  times,  and  the  probable  error  of  the  mean 
is  0.016  feet.  How  many  additional  measurements  of  the  same  pre- 
cision are  necessary  in  order  that  the  probable  error  of  the  mean  shall 
be  only  0.004  feet? 

3.  An  angle  is  measured  by  a  theodolite  and  by  a  transit  with  the 
following  results  : 

By  theodolite 24°  13'  36"  ±    3".! 

By  transit 24    13  24    ±  13  .8 

Find  the  most  probable  value  of  the  angle  and  its  probable  error. 


lOO      DIRECT  OBSERVATIONS   ON  A    SINGLE   QUANTITY.         V. 

4.  A  base-line  is  measured  five  times  with  a  steel  tape  reading  to 
hundredths  of  a  foot,  and  also  five  times  with  a  chain  reading  to  tenths 
of  a  foot,  with  the  following  results  :  — 

By  the  tape  :  741.17  feet.  By  the  chain  :  741.2  feet. 

741.09  feet.  741.4  feet. 
741.22  feet.  741.0  feet. 
741.12  feet.  741-3  feet. 

741.10  feet.  741-1  feet. 

Find  the  probable  errors  and  weights  for  a  single  observation  in  the 
two  cases,  and  also  the  adjusted  length  of  the  line. 

Ans.   741.146  ±  0.012. 

5.  Eight  observations  of  a  quantity  give  the  results  769,  768,  767, 
766,  765,  764,  763,  and  762,  whose  relative  weights  are  i,  3,  5,  7,  8,  6, 
4,  and  2.  What  is  the  probable  error  of  the  general  mean,  and  the 
probable  error  of  each  observation  ? 

6.  The  length  of  a  line  is  stated  by  one  party  as  683.4  ±  0.3,  and 
by  a  second  party  as  684.9  ^  °'3'  What  is  to  be  inferred  from  the  two 
results  ? 


»     J  '  .  » 


91.  LIJVEAR  MEASUKEMENJF^:,      \  ;     '    ';.,■'.•,:  l0'i<, 


\ 


CHAPTER   VI. 

FUNCTIONS   OF   OBSERVED   QUANTITIES. 

90.  In  this  chapter  will  be  discussed  the  determination  of 
the  precision  of  quantities  which  are  computed  from  other 
measured  quantities.  For  instance,  the  area  of  a  field  is  a  func- 
tion of  its  sides  and  angles  :  when  the  most  probable  values 
of  these  have  been  found  by  measurement,  the  most  probable 
value  of  the  area  is  computed  by  the  rules  of  geometry,  and 
the  precision  of  that  area  will  depend  upon  the  precision  of  the 
measured  quantities.  Linear  measurements  will  first  receive 
attention  ;  for,  although  they  are  direct  observations  when  the 
result  alone  is  considered,  yet  really  the  length  of  a  line  is 
a  function  of  its  several  parts,  namely  the  sum.  So,  too,  an 
observed  value  of  an  angle  is  a  function  (the  difference)  of  two 
readings.  All  the  following  reasoning  is  based  upon  the  laws 
of  propagation  of  error  deduced  in  Arts.  68-71. 

Linear  Measurements. 

91.  As  a  line  is  measured  by  the  continued  application  of 
a  unit  of  measure,  its  probable  error  should  increase  with  its 
length.  The  law  of  this  increase  is  given  by  formula  (26).  If 
the  parts  are  all  equal,  and  each  be  taken  as  the  unit  of  length, 
the  number  of  parts  is  the  same  as  the  length  of  the  line.  Let 
r  denote  the  probable  error  of  a  measurement  a  unit  in  length, 
R  the  probable  error  of  the  total  observed  length,  and  /  that 
observed  length.      Then  (26)  reduces  to 

(37)  R=r^l; 


.-lO^    .  ^  .    •  .  FCr/VCT/OjVS   OF  OBSERVED    QUANTITIES.  VI. 

that  is,  the  probable  error  of  a  measurement  of  a  line  increases 
with  the  square  root  of  its  length. 

For  example,  the  value  of  r  for  measurements  with  an 
engineer's  tape  on  smooth  ground  is  about  0.005  :  hence, 
for  a  line  100  feet  long,  R  is  0.05  feet,  and  for  a  line  1,000  feet 
long,  R  is  0.16  feet. 

Since,  by  (18),  weights  are  inversely  as  the  squares  of  probable 
errors,  and,  by  (37),  the  squares  of  probable  errors  are  directly 
as  the  lengths  of  lines,  it  follows  that  the  weights  of  linear 
measurements  are  inversely  as  their  lengths,  or 

(38)  A:A:;>::i:l:i. 

Hence,  if  the  weight  of  a  measurement  of  a  unit's  length  be  i, 
the  weight  of  a  measurement  of  the  length  /will  be  -.     This 

pririciple  is  to  be  used  in  combining  linear  measurements  for 
which  the  value  of  r  is  the  same. 

92.  The  value  of  r  may  be  found  by  measuring  a  line  of  the 
length  /  many  times,  and  computing  R  by  the  methods  of 
the  last  chapter.  Then,  by  (37),  the  value  of  r  is  known.  For 
instance,  take  the  eight  measurements  of  a  line  about  189 
meters  long,  which  are  discussed  in  Art.  84,  for  which  the  proba- 
ble error  of  a  single  observation  was  found  to  be  about  0.05 

meters.     Here  R  =  0.05,  and  then  r  =     ' =  0.004  meters, 

V189 

which  is  the  probable  error  of  a  measurement  of  a  line  one 

meter  in  length. 

The  most  convenient  way,  however,  of  finding  r,  is  to  make 
duplicate  measurements  of  several  lines  of  different  lengths. 
Let  the  lengths  of  the  lines  be  /„  4  .  .  .  /„,  the  differences 
of  the  duplicate  measurements  be  </„  d-.  .  .  .  d„,  and  the  num- 


92. 


LINEAR  MEASUREMENTS. 


103 


ber  of  lines  be  ;/.  These  differences  are  the  true  errors  of  a 
quantity  whose  true  value  is  zero,  and  by  Art.  6"]  the  probable 
error  of  an  observed  difference  is 


r'  =  0.6745  y/ 


'%piP 


n 


Now,  from  Art.  6%,  this  probable  error  is  also 


-'  =  y/r^  4- 


;V2, 


and,  by  equating  these  two  values  of  /-',  it  is  easy  to  find 


(39) 


r=  0.47691/-^': 


which   is  the  probable  error  of   a  measurement  a  unit  long. 
The  weight  /  is  to  be  taken  as  —  in  accordance  with  (38). 

For  example,  the  following  duplicate  measurements  of  the 
sides  of  a  mountain  field,  made  with  a  Gunter's  chain,  may  be 
considered. 


No.  of  Side. 

By  First  Partv. 

By  Second  Party. 

I 

17.21  chains. 

17.18  chains. 

2 

348       " 

3-52        " 

3 

I5-M 

15.19       " 

4 

1.27       " 

1.25        " 

5 

20.06       " 

20.12       " 

6 

8.85       " 

8.92 

7 

0.70       " 

0.70       " 

8 

6.75       " 

6.78       " 

I04  FUNCTIONS   OF  OBSERVED    QUANT/TIES.  VL 

Here  for  the  first  line 

d^  =  0.03,     d^^  —  0.0009,    /i^i^  =  7~  =  0.0000523, 

and  similarly  for  each  of  the  other  lines.  Then,  by  addition, 
^pd"-  =  0.001855,  and  lastly,  from  (39),  the  probable  error  of  a 
measurement  of  a  unit's  length  (that  is,  of  one  chain)  is  0.0073 
chains,  or  0.73  links. 

93.  The  general  formula  (26)  shows  clearly  how  the  pre- 
cision of  linear  measurements  depends  upon  the  precision  of 
the  parts.  Evidently  the  fewer  the  parts,  the  smaller  will  be 
7?,  and  the  greater  the  precision.  Also  the  longer  the  chain, 
the  fewer  will  be  the  parts,  and  the  greater  the  precision. 

It  must  be  carefully  noted,  however,  that  the  preceding  rea- 
soning only  applies  to  the  accidental  errors  (Art.  7)  of  obser- 
vation, and  that  all  constant  errors  must  be  investigated,  and 
removed  from  the  results,  before  the  formulas  {^y)  and  (39)  are 
used.  The  effects  of  temperature  on  the  length  of  the  chain 
or  tape,  for  instance,  may  be  removed  by  reading  the  ther- 
mometer, and  applying  the  proper  computed  corrections,  and 
the  effects  of  side  deviations  may  be  removed  by  making  the 
chain  sufficiently  longer  at  the  start.  In  general,  the  constant 
errors  of  linear  measurements  increase  directly  as  the  length 
of  the  line  ;  while  only  the  accidental  errors  increase  as  the 
square  root  of  the  length. 

Angle  Measurements. 

94.  The  measurement  of  an  angle  is  in  general  effected  by 
taking  the  difference  of  readings  from  a  graduated  limb  ;  and 
these  readings,  in  their  turn,  may  be  the  means  of  readings  on 
two  or  more  verniers.  By  the  use  of  the  principle  expressed 
in  formula  (25)  it  is  possible  to  determine  the  precision  of 
these  readings  from  the  probable  errors  of  observed  results. 


§94- 


ANGLE  MEASUREMENTS. 


105 


As  an  example,  the  following  measurements  of  an  angle  made 
with  a  transit  having  two  verniers  reading  to  minutes  will  be 
discussed.  The  angle  was  chosen  at  about  35°  in  order  that 
eleven  readings  might  approximately  go  around  the  circle,  and 
each  reading  is  the  mean  of  the  two  verniers. 


On  Vernier  A. 


5  03  30 
39  59  30 
74  55  00 
109  49  30 
144  45  00 
I  79  41  00 
214  37  00 
249  32  30 
284  28  00 
319  24  00 

354   19  30 


On  Vernier  B. 


O  I  II 

5  03  30 
39  60  00 

74  55  30 

109  50  GO 

144  45  GO 

179  41  00 

214  36  30 

249  32  GO 

284  27  30 

319  23  30 

354   19  30 


Mean  Reading. 


O  I  II 

5  03  30 

39  59  45 

74  55   15 
109  49  45 

144  45  00 

179  41  00 

214  36  45 

249  32   15 

284  27  45 

319  23  45 
354   19  30 


Angle. 


34°  56' 15" 
55  30 

54  30 

55  15 

56  00 

55  45 
55  30 
55  30 

56    GG 

55  45 


By  the  method  of  the  last  chapter  it  is  easy  to  find  that  the 
probable  error  of  a  single  observation  of  an  angle  is  nearly  20". 
Let  r,  represent  the  probable  error  of  a  reading  on  one  vernier, 
and  r^  that  of  the  mean  of  the  two  verniers.  Then  by  (25), 
since  each  observation  is  the  difference  of  two  readings, 


2G 


^fV-\-f'2-,     or  r^  =  14".!. 


Next  for  r,  the  formula  (19)  gives 


14. 1  =■---, 
V2 


or  ;-,  =  20 


So  it  appears  that  the  probable  error  of  a  single  observation  of 
an  angle  taken  in  the  above  manner  is  the  same  as  that  of  a 


I06  FUNCTIONS   OF  OBSERVED   QUANTITIES.  VI. 

single  reading  on  one  vernier.  The  reading  of  both  verniers 
not  only  eliminates  the  error  of  eccentricity,  but  adds  much  to 
the  precision  of  the  results. 

95.  By  the  method  of  repetitions  the  precision  of  angle 
measures  can  be  further  increased.  The  observations  should 
be  conducted  like  those  above  described,  except  that  the  plate 
is  turned  n  times  between  the  two  readings.  Let  i\  be  the 
probable  error  of  a  mean  reading,  and  }\  that  of  the  observed 

result,  which   is  -th  of   the    difference    of   the    two    readings. 
11 

Then  by  (25)  and  (27),  neglectmg  the  error  in  pointing, 

r.—  —  V2. 
^         II 

By  the  method  of  the  last  article  the  mean  of  ;/  readings 
would  give 

Sn 

The  precision  of  ;/  repetitions  is,  hence,  \fi  times  greater  than 
the  mean  of  n  independent  observations.  However,  the  errors 
in  pointing,  and  other  causes,  render  it  doubtful  if  it  is  ever 
aavantageous  to  make  u  exceed  six  or  eight. 


"■to^ 


Precision  of  Areas. 

96.  Let  z^  and  z^  be  the  measured  sides  of  a  rectangle,  and 
r,  and  r^  their  probable  errors.  Then  by  (29)  the  probable 
error  of  the  computed  area  z^z-,  is 

If  r  be  the  probable  error  of  a  measurement  a  unit  in  length, 
the  law  of  (37)  gives 

r,2  =  r^Zy     and    r,^  =  r^z,.  \ 


§  98.  REMARKS  AND  PROBLEMS.  lO/ 

and  hence  the  probable  error  in  the  area  is 


For  instance,  let  a  lot  60  X  150  feet  be  laid  out  by  an  en- 
gineer's chain,  for  which  r  =  0.01.  Then,  by  the  formula, 
-^  =  13-75  square  feet,  which  is  the  probable  error  of  9,000 
square  feet,  the  computed  area. 

97.  By  the  application  of  formula  (30)  the  probable  error 
of  any  computed  area  can  be  found  from  the  known  probable 
errors  of  its  sides  and  angles.  As  one  of  the  simplest  cases, 
take  a  triangle  ABC,  whose  area  is  found  from  the  angle  A  and 
the  two  adjacent  sides  AB  and  AC.     The  observed  values  are 

AB  =  252.52  ±  0.06, 
AC  =  300.01  ±  0.06, 
A     =  42°  13' 00"  ±  30". 

The  area  of  this  triangle  is  ^-AB.AC.sin  A  =  25,453  square  feet. 
To  compare  with  (30)  let  AB  =  .o-,,  AC  =  ^2.  and  sin  A  =  ;:^; 
also  r,  :=  r^  =  0.06,  and  r^  =  o.oooii  =  tabular  difference  corre- 
sponding to  30". 


en 

dZ 
dzi 

\A  C.  sin  A, 

dZ 

dz^ 

\AB.  sin  A, 

dZ 

dz. 

\  A  B.AC. 

By  inserting  all  values  in  (30)  it  is  easy  to  find  R  =  8.9  square 
feet  for  the  probable  error  of  the  area. 


Remarks  and  Problems. 

98.  By  the  application  of  formulas  (25)  to  (30)  the  precision 
of  many  other   functions    of   observed    quantities   than   those 


IC8  FUNCTIONS  OF  OBSERVED    QUANTITIES.  VI. 

above   noticed  may  be  investigated.     A    few  of   the  simplest 
cases  are  included  among  the  following  problems. 

1.  The  radius  of  a  circle  is  observed  as  looo  ±  0.2.  Find  the 
probable  errors  of  its  circumference  and  area. 

2.  Find  the  maximum  probable  error  of  sin  A  +  cos  A  when  the 
probable  error  of  A  is  20". 

3.  In  order  to  determine  the  difference  of  level  between  two  points 
A  and  B,  an  instrument  was  set  up  halfway  between  them,  and  twenty 
readings  taken  on  rods  held  at  each  point,  with  the  following  results  : 

Rod  at  A.  Rod  at  B. 

7  readings  gave  7.229  feet.  3  readings  gave  9.806  feet. 

8  readings  gave  7.230  feet.  12  readings  gave  9.807  feet. 
5  readings  gave  7.231  feet.  5  readings  gave  9,808  feet. 

What  is  the  most  probable  difference  of  level  between  the  two  points 

and  the  probable  error  of  the  determination? 

Ans.  2.5772  ±0.00015. 

4.  k  block  of  cast-iron  weighing  100  pounds  rests  upon  a  horizontal 
table,  also  of  cast-iron.  A  horizontal  force  is  applied  to  the  block,  and 
it  is  obser\-ed  that  it  begins  to  move  when  the  force  is  15.5  pounds.  If 
the  probable  error  in  the  determination  of  this  force  is  0.5  pound,  what 
is  the  probable  error  of  the  co-efficient  of  friction  ? 

5.  A  chronometer  is  rated  at  a  certain  date,  and  found  to  be  9"'  12^.3 
fast,  with  a  probable  error  of  0^.3.  Ten  days  afterwards  it  is  again  rated, 
and  found  to  be  9"' 21^.4  fast,  with  the  same  probable  error.  What  is 
the  probable  error  of  the  mean  daily  rate  ? 

6.  A  line  of  levels  is  run  in  the  following  manner :  the  back  and  fore 
sights  are  taken  at  distances  of  about  200  feet,  so  that  there  are  thirteen 
stations  per  mile,  and  at  each  sight  the  rod  is  read  three  times.  If  the 
probable  error  of  a  single  reading  is  0.00 1  feet,  what  is  the  probable 
error  of  the  difference  of  level  of  two  points  which  are  ten  miles  apart? 


§99-  METHOD   OF  PROCEDURE.  IO9 


CHAPTER    VII. 

INDEPENDENT   OBSERVATIONS    ON    SEVERAL   QUANTITIES. 

99.  Independent  observations  on  several  related  quantities 
are  to  be-  adjusted  by  the  methods  of  Arts.  46-50,  and  thrir 
precision  determined  by  the  methods  of  Arts.  72-76.  Tiie 
following  are  the  steps  of  the  process  : 

1st,  Let  .s",,  ^•j,  £-3,  etc.,  represent  the  quantities  to  be  det^^r- 
mined,  and  for  each  observation  write  an  observation  equation  ; 
or,  if  more  convenient,  let  z„  z^,  z^,  etc.,  be  corrections  to 
assumed  approximate  values  of  the  unknown  quantities. 

2d,  From  the  observation  equations  form  the  normal  equa- 
tions, which  will  be  as  many  as  there  are  unknown  quantities. 

3d,  Solve  the  normal  equations  :  the  resulting  values  of  the 
unknown  quantities  will  be  their  most  probable  values,  that  is, 
the  best  values  that  can  be  deduced  from  the  given  observations. 

4th,  Find  the  residuals,  and  the  probable  error  of  an  obser- 
vation of  the  weight  unity  from  formula  (32). 

5th,  Find,  if  desired,  the  weights  and  probable  errors  of  the 
adjusted  values  of  the  unknown  quantities. 

When  the  number  of  unknown  quantities  exceeds  four  or 
five,  it  will  usually  be  found  most  convenient  to  use  the  algo- 
rithm of  formulas  (10)  and  (i  i)  for  observations  of  equal  weight, 
and  of  (12)  and  (13)  for  those  of  unequal  weight,  and  to  solve 
the  normal  equations  by  the  method  of  Arts.  51-55.  It  will, 
however,  probably  be  best  for  a  beginner  to  form  the  normal 


no  INDEPENDENT  OBSEKVATIOAS.  VII. 

equations  by  the  rules  in  Art.  48   and   Art.  50,  and   to  solve 
them  by  his  own  algebraic  method. 

It  will  often  be  convenient  to  take  the  unknown  quantities 
as  corrections,  rather  than  as  the  real  quantities  themselves; 
since  thus  the  numbers  entering  the  computation  are  much 
smaller.  The  following  practical  examples  will  illustrate  the 
whole  method  of  procedure. 

Discussion  of  Level  Lines. 

100.  The  following  observations  are  recorded  in  the  Report  of 
the  United  States  Geological  and  Geographical  Survey  for  1873, 
and  are  here  supposed  to  be  of  equal  reliability  or  weight : 

1.  Z,  above  6>,  573.08  feet,  by  Coast    Survey   and    canal    levels,   via 

New  York  and  Albany. 

2.  Z,  above  Z,,      2.60  feet,  by  observations  on  surface  of  Lake  Erie. 

3.  Z2  above  O,  575.27  feet,  by  Coast  Survey  and  railroad  levels,  via 

New  York  and  Albany. 

4.  Z3  above  Z,,  167.33  feet,  by  railroad  levels. 

5.  Zj above  Z3,      3.80  feet,  by  railroad  levels. 

6.  Z,  above  Z,,  170.28  feet,  by  railroad  levels,  via  Alliance. 

7.  Zj  above  Z5,  425.00  feet,  by  railroad  levels. 

8.  Zj  above  O,  319.91  feet,  by  railroad  and   Coast  Survey  levels,  via 

Philadelphia. 

9.  Z5  above  O,  319.75  feet,  by  railroad  levels,  via  Baltimore. 

The  letters  here  have  the  following;  meanings  : 


O 
Z. 

z, 
z. 


s  the  mean  surface  of  the  Atlantic  Ocean. 
s  the  mean  surface  of  Lake  Erie  at  Buffalo. 
s  Cleveland  city  datum  plane. 
s  Depot  track  at  Columbus,  O. 
s  Union  Depot  track  at  Pittsburg. 
s  Depot  track  at  Harrisburg. 


§  lOO.  D/SCUSS/ON  OF  LEVEL   LINES.  Ill 

It   is   required   to  adjust   these   observations,  and  to  find  the 
probable  error  of  a  single  observation. 

1st,  Represent  the  unknown  heights  of  Z,,  Z^,  Z^  Z^,  and  Z^ 
by  ^„  -Cj,  Tj,  z^,  and  ^5.  Then  the  observations  give  the  obser- 
vation equations 

2.  =  5  73-o8> 

S2  —  2i  =         2.60, 
23  =   575.27, 

z^-z^=  167.33, 

^4  -  -3  =      3-8o» 
S4  —  s,  =  170.28, 

S4  -  -5   =    425-00. 

25  =  3i9-9i» 
25  =  3i9-75- 

2d,  Form  a  normal  equation  for  z,  by  multiplying  each  equa 
tion  in  which  z^  occurs  by  its  co-efBcient  in  that  equation,  and 
adding  the  products  ;  and  in  the  same  way  form  a  normal  equa- 
tion for  each  of  the  other  unknown  quantities.     This  gives 

22,  —    S2  =5  70.48, 

—  s,  +  4S2  —23—24  =  240.26, 

—  2,  +  2S3  -    24  =  163.53, 

-  Z^-     23  4-  ZZ,  -     25  =  599.08, 

—      ^4  +  3^5  =   214.66. 

3d,  The  solution   of  these  normal  equations  furnishes  the 
following  values :  — 

2i  =  572-81,         S2  =  575-14,  23  ~  742.05, 

24  =  745-43,         25  =  320.03, 

which    arc   the   adjusted   elevations   of   the    five   points  above 
the  datum  O. 


I  12 


INDEPEA'DENT  OBSERVATIONS. 


VII. 


4th,  Substitute  these  values   in   the   observation   equations, 
and  find  the  residuals  and  their  squares  ;   thus  : 


No. 

V. 

.z/^. 

I 

0.27 

0.073 

2 

•27 

•073 

3 

•13 

.017 

4 

.42 

.176 

5 

.42 

.176 

6 

.01 

.000 

7 

.40 

.160 

8 

.12 

.014 

9 

.28 

.078 

2z'^ 

=  0.767 

Here  the  number  //  of  observations  is  9,  and  the  number  q  ol 
unknown  quantities  is  5.  The  weights  /  are  all  unity.  Then, 
from  {12), 


=  0.67451/^1^  =  0.295  feet, 


which  is  the  probable  error  of  an  observation  of  weight  unity. 

5th,  To  determine  the  probable  errors  of  the  above  adjusted 
values,  it  is  necessary  to  find  their  weights  by  the  method  of 
Art.  75.  For  instance,  to  find  the  weight  of  z^,  represent  the 
absolute  term  in  the  normal  equation  for  z^  by  B,  and  put  all 
the  other  absolute  terms  equal  to  zero.  Then  the  solution 
gives  z^  =  \B,  and  accordingly  the  weight  of  z^  is  26-  Hence 
the  probable  error  of  the  value  of  z^  is 

0.295 


v/1.96 


=  o.  21 1  feet; 


§  lOI.  DISCUSSION-  OF  LEVEL    LINES.  II3 

SO  that  the  final  elevation  of  Z^  may  be  written 

^2  =   575-14   ±  0-2I, 

and  it  is  an  even  wager  that  the  actual  error  in  the  value  575.14 
is  less  (or  greater)  than  the  amount  0.21  feet. 

loi.  For  level  lines  of  unequal  precision  the  process  of  ad- 
justment is  the  same,  except,  that,  before  forming  the  normal 
equations,  each  observation  equation  should  be  multiplied  by 
the  square  root  of  its  weight.  To  illustrate,  regard  the  above 
nine  observations  as  of  unequal  weight.  The  least  trustworthy 
is  No.  9 ;  because  it  is  not  known  that  mean  tide  at  Baltimore 
is  the  same  as  the  mean  surface  of  the  ocean,  and  its  weight 
may  be  taken  as  i.  Nos.  3  to  8  inclusive  are  ordinary  railroad 
levels,  and  may,  with  reference  to  No.  9,  be  given  a  weight  of  4. 
Nos.  I  and  2,  being  the  result  of  carefully  conducted  govern- 
ment and  canal  levels  extending  over  many  years,  are  the  most 
reliable  of  all ;  and  a  weight  of  25  may  be  assigned  them.  The 
observation  equations  are  the  same  as  before ;  multiplying  each 
by  the  square  root  of  its  weight  gives 

52,  =  2S65.40, 
522  —  521=      13-00. 

203  =  1150-54, 
2S3  -  22,  =  334-66, 
224  -  223  =        7-6o, 

2^4  —  22^  =      340.56, 

224  —  225  =    850.00, 

225  =   639.82, 

h  =    3I9-75- 
The  normal  equations  now  are 

5021  —  2522  =  14262.00, 

■  2521  -f  3722  —  423  —  424             =  1015.64, 

—  422  +  823—  424             =  654.12, 

-  422  -  423  +  1224  -  425  =  2396.32, 

—    424  +  925  =  —100.61, 


114 


INDEPENDENT  OBSERVATIONS. 


VII. 


and  their  solution  sives 


2,  =  572.98,        2^  =  575.48,        23=742.36, 


745-72, 


320.25. 


Inserting  these  in  the  observation  equations,  the  remainders 
or  residuals  t^,,  c'^,  etc.,  are  found,  and  placed  in  the  third  column 
below,  their  squares  in  the  fourth,  and  the  product  of  each 
square  by  its  corresponding  weight  in  the  fifth. 


No. 

P- 

w. 

z/^ 

pzF. 

I 

25 

O.IO 

O.OIO 

0.250 

2 

25 

.1 1 

.012 

.300 

3 

4 

.20 

.040 

.160 

4 

4 

•44 

.194 

.776 

5 

4 

•43 

.185 

.720 

6 

4 

.02 

.000 

.002 

7 

4 

.48 

.210 

.840 

8 

4 

•34 

.116 

.464 

9 

I 

•50 

.250 

.250 

-%pv^  -. 

=  3^762 

Then  by  (32)  the  probable  error  of  an  observation  of  weight 
unity,  that  is  of  No.  9,  is 


=  0.6745^/^^^^  =  0.635  feet, 


and  the  probable  error  of  observations  i  and  2  is  by  (31) 


—^A  =  0.13  feet, 
5 

0.635  _ 


and  of  those  from  3  to  8  inclusive  is 


=  0.32  feet. 


§  I02.  DISCUSSION  OF  LEVEL   LINES.  II5 

In  order,  lastly,  to  find  the  probable  errors  of  the  above 
adjusted  values,  their  weights  must  be  determined.  For  in- 
stance, to  find  the  weight  of  ^^,  place  the  absolute  term  in 
the  fourth  normal  equation  equal  to  A,  and  those  in  the  other 
normal  equations  equal  to  zero.  Then  the  solution  gives 
^4  =  ||^.4,  and  accordingly  the  weight  of  z^  is  6.62.  Hence 
the  probable  error  of  the  value  of  z^  is 

0-635 

Tz,  =  ", =  0.2=;  feet. 

*       V6.62  ^ 

And  in  a  similar  way  the  probable  error  of  the  value  of  z^  may 
be  found  to  be  0.15  feet. 

102.  For  such  sim.ple  cases  as  those  just  presented,  the  abso- 
lute terms  in  the  normal  equations  might  be  represented  by 
letters,  A„  A^,  etc.,  and  a  general  solution  easily  effected,  which 
would  give  at  once  all  the  weights  and  unknown  quantities. 
For  instance,  if  the  normal  equations  of  Art.  100  are  thus 
written 

2S.  —     Z2  =  A„ 

2_|_1-    ^    y  —     J 

-1  ^  4-^2  -3  ■^4  —  ^23 

-  s,  +  223  -     S4       ■        =  ^3, 

-  2,  -      S3  +  32,  -      25  =   ^4, 

the  solution  gives 

51S.  =  32^,  +  13^2  +  ii^^3  +    9.-/4  +  3^5, 

5123   =    13.4,   +   26^2  +   22^3   +    18^4  +  6^5, 

5123  =  II  A,  +  22A._  +  50^3  +  27-44  +  9^5, 
1724=  3^.+  6^3+  9^3+12.44  +  4^5, 
1725=      A,+    2^3+    3^3+    4^4+7^5, 

where  all  the  weights  are  at  once  seen,  and  from  which  the 
values  of  the  unknown  quantities  can  easily  be  found. 


Il6  INDEPENDENT  OBSERVATIONS.  VII. 

As  indicated  in  Art.  99,  the  numerical  operations  may  be 
somewhat  simplified  by  taking  the  unknown  quantities  as  cor- 
rections to  be  applied  to  assumed  elevations  of  Z„  Z^,  etc. 
Thus  it  is  seen  from  the  observations  that  573  and  575  feet  are 
approximate  elevations  for  Z,  and  Z^.     By  writing,  then, 

elevation  of  Z,  =  573  +  Sj, 
elevation  of  Z,  =  575  -\-  z^, 
elevation  of  Z,  =  742  +  z^, 
elevation  of  Z^  =  745  +  z^, 
elevation  of  Z5  =  320  +  z^, 

the  following  simpler  observation  equations  are  obtained  from 

the  given  data : 

2,  =  0.08, 

2,  —  2,  =  0.60, 

z^  =  0.27, 

^3  —  Z2   =  0.33, 

0^  —  Zj  =        0.80, 

2^  —  ^2  =  0.28, 

Z^  —  ^5   =  0.00, 

S5   =    —   0.09, 

From  these  the  normal  equations  are  formed,  whose  first  mem- 
bers are  the  same  as  written  above,  and  whose  second  mem- 
bers have  the  values /i,  =— 0.52,  A^  =  -]r0.26,  A^=  —0.47, 
A^=^  -\-  1.08,  A.=^  —  0.34.  The  solution  of  the  normal  equa- 
tions gives 

z,=  —  0.19,     z^  =  0.14,     2,  =  0.05,     z^  =  0.43,     25  =  0.03  ; 

and  the  final  elevations  are 

^.  =  573-00  —  0-19  =  572.81, 

^2  =  5  75-00  +  0.14  =  5  75-14,     etc., 

which  are  the  same  as  found  in  Art.  100. 


§103.  ANGLES  AT  A  station:  II7 


Angles  at  a  Station. 

103.  When  two  angles  and  also  their  sum  are  observed  at  a 
station,  the  observed  sum  usually  differs  from  the  sum  of  the 
two  measured  single  angles.  Let  the  observation  of  the  first 
angle  give  the  result  M„  of  the  second  M^,  and  that  of  their 
sum  My  Then  J/,  +  iJ/,  is  greater  or  less  than  M3  by  a  cer 
tain  discrepancy  d.  It  is  required  to  adjust  the  observations, 
regarding  the  weights  as  equal,  and  to  find  the  probable  errors 
of  the  adjusted  values. 

1st,  Let  ^,  and  z.^  be  the  most  probable  corrections  to  the 
observed  values  J/,  and  J/,,  so  that  J/,  +  z^  and  M^  +  ^2  are 
the  most  probable  values  of  the  first  and  second  angles.  The 
observation  equations  then  are 

J/,  +  s,    =  Af„ 
M^  +  z,-  =Af„ 

which  reduce  to 

Zi  =  o, 
Z2  =  o, 
z,  +  z,  =  J/3  -  (Af,  +  Af,)  =  (i. 

2d,  From  these,  the  normal  equations  are 

22,  +    22  =  d, 
2,  +  22,  =  d. 

3d,  The  solution  of  the  normal  equations  gives 

2,  =  ^d,     and  23  =  -^d, 

for   the    most   probable   values  of  the  corrections :    hence  the 
adjusted  values  are 

M.  +  Id, 

M,  +  \d, 

■     M3  -  \d. 


Ii8 


INDEPENDENT  OBSERVATIONS. 


VII. 


4th,  The  residuals  are  evidently  the  three  corrections,  the 
sum  of  whose  squares  is  \d^ ;  then,  from  (32), 

r=  0.6  745  v^^r/^  =  0.389^, 

which  is  the  probable  error  of  a  single  observed  value. 

5th,  By  the  method  of  Art.  75  it  is  easy  to  find  that  the 
weights  of  the  adjusted  values  of  ^,  and  z^  are  1.5  :  hence 
their  probable  errors  are 


0.389^ 
1^ 


=  0.318^, 


and   evidently   the   probable    error   of   the   adjusted   value   of 
rr, -|- -2  is  also  0.318^^^ 

104.  When  several  angles  are  observed  at  a  station,  several 
sums  and  differences  of  simple  angles  are  often  taken.  For 
example,  the  following  are  the  angles  observed  at  the  Station 
Hillsdale,  on  the  United  States  Lake  Survey  ;  each  being  the 
mean  of  nearly  the  same  number  of  readings,  and  hence  re- 
garded as  of  the  same  weight.  (See  Report  of  United  States 
Lake  Survey,  p.  449.) 


No. 

Between  Stations. 

Observation. 

I 

Bunday  and  Wheatland 

44°    25' 

4o".6i3 

2 

Bunday  and  Pittsford 

80     47 

32.819 

3 

Wheatland  and  Pittsford 

36     21 

51.996 

4 

Pittsford  and  Reading 

91      34 

24.758 

5 

Pittsford  and  Bunday 

279      12 

27.619 

6 

Reading  and  Quincy 

62     37 

43405 

7 

Quincy  and  Bunday 

125        GO 

18.808 

§i04. 


ANGLES  AT  A   STATION. 


119 


The  annexed  figure  shows  the  relative  positions  of  the  sta- 
tions and  of  the  seven  observed  angles.  It  is  required  to 
adjust  the  observed  results,  and  to  find  their  probable  errors. 


Bunda 


Pittsford 


1st,  Let  Z^,  Z3,  Z^,  and  Z^,  be  the  required  most  probable 
values  of  four  of  the  simple  angles  as  indicated  in  Fig.  7; 
then  the  observation  equations  are 


360°- 


Z,  =  44° 

25' 

4o".6i3, 

Z,  +  Z3  =.  80 

47 

32.819, 

Z3  =  36 

21 

51.996, 

Zj  =  91 

34 

24-758, 

36o°-(Z.+Z3)  =  2  79 

12 

27.619, 

Ze  =  62 

37 

43-405» 

{Z,  +  Z,+    Z,^Z,)  =  125 

GO 

18.808. 

Assume  the  measured  values  of  Z„  Z^  Z^,  and  Zg  as  approxi- 
mate,  and  let  ^„  z^  z^,  and  Zf,  be  the  most  probable  corrections, 


thus 


Z,  =  44° 

^3  =  36 
Z,  =  91 

Z^=  62 


25'  4o".6i3  +  2., 
21  51.996 +Z3, 
34  24.758  +  24, 
37  43  405  +  Zf,. 


I20  INDEPENDENT  OBSERVATIONS.  VII. 

Then,  by  inserting  these  values  in  the  observation  equations, 
the  following  simpler  observation  equations  are  found  : 


2, 



0, 

2. 

24 

= 

+ 

0.210, 
0, 

z, 

+  Z3 

Z6 





0.228, 
0, 

z. 

+ 

h 

+ 

24 

+  Z6 

= 

+ 

0.420, 

in  which  the  right-hand  members  denote  seconds  only. 

2d,  The  normal  equations  are  now  easily  written,  either  by 
the  rule  of  Art.  48,  or  by  the  help  of  the  algorithm  of  formulas 
(10)  and  (11).     They  are 

4^1  +  3^3  +    2:4  +    26  =  +  0.402, 

IZ,  +  423  +  2^  +  26  =  +  0.402, 
2,  +  Z^-\r  2Z^  4-  26  =  +  0.420, 
2,  +      23  +     Z^+  226  =    +  0.420. 

3d,  The  solution  of  these  equations  gives 

2,  =  23  =    +  0".022,  2^  =  26  =  o".I26. 

The  addition  of  these  corrections  to  the  approximate  values 
gives  the  most  probable  values  of  the  angles  Nos.  i,  3,  4,  and  6; 
and  from  these,  by  simple  addition,  the  most  probable  values  of 
Nos.  2,  5,  and  7,  are  obtained.     Thus,  the  adjusted  values  are 


No.  I  =  44° 

25' 

4o".635  =  Z„ 

No.  3  =  36 

21 

52.018  =  Z3, 

No.  4  =  91 

34 

24.884  =  Z„ 

No.  6  =  62 

37 

43.531  =  ^, 

No.  2  =  80 

47 

32.653  =  Z.  +  Zj, 

No.  5  =  279 

12 

27-347  =  360°-  (Z. +  Z3), 

No.  7  =  125 

00 

18.932  =  360°  -  (Z.  +  Z,-hZ,-i-  Ze). 

^i04. 


ANGLES  AT  A    STATION. 


121 


4th,  The  differences  between  the  observed  and  the  adjusted 
values  are  the  residuals,  which,  with  their  squares,  are  thus 
arranged  : 


No. 

Observed. 

Adjusted. 

V. 

v". 

I 

4o".6i3 

4o".635 

+  0022 

0.0005 

2 

32.819 

32-653 

—  0.166 

^      .0276 

3 

51.996 

52.018 

+  0.022 

.0005 

4 

24.758 

24.884 

+  0.126 

.0159 

5 

27.619 

27-347 

—  0.272 

.0740 

6 

43405 

43-531 

+  0.126 

.0159 

7 

18.808 

18.932 

+  0.124 

.0154 

The  sum  ^v^  is  here  0.1498;  and  hence,  by  formula  (32),  the 
probable  error  of  a  single  observation  is 


,         /0.1498        „ 
r=  o.6745y — ^^  =  o  .15] 


5th,  By  writing  A  for  the  absolute  term  in  the  first  normal 
equation,  and  zero  for  the  absolute  terms  in  the  other  nor- 
mal equations,  the  solution  gives  the  value  of  z^  as  ^-^A  ;  and 
hence  the  weight  of  z,  is  1.7.  In  a  similar  way  the  weight  of 
z^  is  found  to  be  1.4.  The  probable  errors  of  the  adjusted 
values  of  z^  and  z^  are  now 


0-151 

Vi-7 


=  o".ii6; 


and  those  of  the  adjusted  values  of  z^  and  z^  are 


122 


INDEPENDENT  OBSER  VA  TIONS. 


VII. 


In  order  to  find  the  probable  errors  of  angles  Nos.  2,  5,  and  7, 
it  would  be  necessary  to  represent  them  by  single  letters,  and 
to  form  and  solve  another  set  of  normal  equations. 

105.  As  an  example  of  angles  with  unequal  weights,  the  fol- 
lowing observations  at  North  Base,  Keweenaw  Point,  on  the 
United  States  Lake  Survey,  will  next  be  considered  : 


No. 

Between  Stations. 

Observed  Angle. 

Weight. 

I 

2 

3 
4 
5 

Crebassa  and  Middle 
Middle  and  Quaquaming 
Crebassa  and  Quaquaming 
Quaquaming  and  South  Base 
Middle  and  South  Base 

55°   5/  58"-68 

48     49     13.64 

104     47     12.66 

54     38     15-53 
103     27     28.99 

3 

19 

17 

13 
6 

Let  Z„  Z2,  and  Z^  represent  the  angles  Nos.   i,  2,  and  4; 
then  the  observation  equations  are 

Z.=    55°  57'58".68,  with  weight  3, 

Z2  =     48  49      13.64,  with  weight  19, 

Z,  +  Zj  =  104  47     12.66,  with  weight  17, 

^4=    54  Z^     15-53.  with  weight  13, 

Zj  +  ^4  =  103  27     28.99,  ^^'^^h  weight  6. 

Let  z^,  z^,  and  z^  be  corrections  to  the  measured  values  of 
Z^,  Z^,  and  Z^\  then  the  simpler  observation  equations  are 


z,  =       o,  with  weight    3, 

2j  =       o,  with  weight  19, 

2.  +  22  =  -f-  0.34,  with  weight  17, 

z^  =       o,  with  weight  13, 

ij  -f  z^  =  —  0.18,  with  weight    6. 


K)5. 


ANGLES  AT  A    STATION. 


123 


From  these,  the  normal  equations  are  formed,  either  by  the 
rule  of  Art.  50,  or  by  the  help  of  the  algorithm  of  formulas 
(12)  and  (13).     They  are 

202,  +  1722  =  +  5.78, 

172, +  4222-1-    624  =-1-4.70, 

62^  +  1924  =  —  1,08. 

The  solution  of  these  equations  gives 

2.  =    -I-  o".285,  2a  =    -f  0".005,  Z^—   —  O  .O59. 

Hence  the  following  are  the  adjusted  angles 

No.  1=  55°  57'58".965, 
No.  2  =  48  49  13.645, 
No.  3  =  104     47     12.610, 

No.  4  =  54  38  15-471, 
No.  5  =  103     27     29.116. 

To  find  the  probable  errors,  the  residuals  are  next  obtained. 


No. 

Observed. 

Adjusted. 

V. 

v^. 

p- 

pv\ 

I 

58".68 

58".965 

+  0.285 

0.0812 

3 

0.244 

2 

13.64 

13-645 

-f  0.005 

.0000 

19 

,000 

3 

12.66 

12.610 

-  0.050 

.0025 

17 

.042 

4 

15-53 

15-471 

-  0.059 

•0035 

13 

•045 

5 
1 

28.99 

29.1 16 

-l-  0.126 

.0159 

6 

•095 

The  sum  ^pv'^  is  here  0.426  ;  then,  by  (32), 


=  o.6745/°4 


26 


^11        ^T 


124  INDEPENDENT  OBSERVATIONS.  VII. 

which  is  the  probable  error  of  an  observation  of  the  weight 
unity.  The  probable  error  of  the  observed  angle,  No.  2,  is, 
then, 

o.  -JI 

^  tr 

r^  =  -7=  =  o  .07. 
V/19 

The  probable  error  of  the  final  value  of  No.  2  must  be  less 
than  o".07,  since  its  weight  is  increased  by  the  adjustment. 


Empirical  Constants. 

106.  One  of  the  most  important  applications  of  the  Method 
of  Least  Squares  is  the  deduction,  from  observations,  of  the 
values  of  physical  constants  or  co-efificients.  In  all  such  cases 
a  theoretical  formula  or  law  is  first  established,  which  contains 
the  co-efficients  in  a  literal  form  ;  and  this  law  serves  to  state 
as  many  observation  equations  as  there  are  observations.  The 
method  of  procedure  is  then  exactly  the  same  as  that  outlined 
in  the  first  article  of  this  chapter.  The  precision  of  the  values 
deduced  for  the  constants  depends,  of  course,  upon  the  precision 
and  number  of  the  observations  which  enter  the  discussion. 

As  an  example,  take  the  determination  of  the  ellipticity 
of  the  earth  by  means  of  experiments  on  the  length  of  the 
seconds'  pendulum.  In  1743  Clairaut  deduced  the  following 
remarkable  law  : 

s  =  S  -^  S{\k  -f)%mH, 

in  which  5  is  the  length  of  the  seconds'  pendulum  at  the 
equator,  and  s  its  length  at  any  latitude  /,  while  k  is  the  ratio 
of  the  centrifugal  force  at  the  equator  to  gravity,  and  f  is  the 
fraction  expressing  the  ellipticity  of  the  earth.  This  may  be 
written 

s  =  S  ■\-  Tsin^/. 


§  io6. 


EMPIRICAL    CONSTANTS. 


I2S 


Now,  by  measuring  s  at  two  different  latitudes,  two  equations 
would  result,  from  which  values  of  6"  and  T  could  be  found  ; 
and,  by  measuring  s  at  many  different  latitudes,  many  equa- 
tions would  result,  from  which  the  most  probable  values  of  5 
and  T  may  be  found.  The  following,  for  instance,  are  thkteen 
observations,  taken  by  Sabine  in  the  years  1822-24: 


Place. 

Latitude. 

Length  of  Seconds' 
Pendulum. 

English  Inches. 

Spitzbergen 

+  79°  49' 58" 

39.21469 

Greenland 

74  32   19 

39-20335 

Hammerfest 

70  40     5 

39-19519 

Drontheim 

63  25  54 

39-17456 

London 

51   31     8 

39-13929 

New  York 

40  42  43 

39.10168 

Jamaica 

17  56     7 

39-03510 

Trinidad 

10  38  56 

39.01884 

Sierra  Leone 

8  29  28 

39.01997 

St.  Thomas 

0  24  41 

39-02074 

Maranham 

-2   31   43 

39.01214 

Ascension 

7  55  48 

39.02410 

Bahia 

12  59  21 

39.02425 

For  each   of  these   an    observation    equation   is   now  to  be 
written.     Thus,  for  the  first. 


s 
I 

sin/ 

sin^/ 

39.21469 


39.21469. 

79   49   58  . 
0.9842965. 

0.9688402. 

>S  -f  0.96884027: 


126  INDEPENDENT  OBSERVATIONS.  VII. 

And  in  like  manner  the  following  thirteen  observation  equations 
are  stated : 

39.21469  =  ,5"  +  0.9688402  Zl 

39-20335  =  S  +  0.92893047: 

39.19519  =  .S  -|-  0.8904120/1 

39.17456  =  ^  +  0.79995447: 

39.13929  =  S  -\-  0.61279667: 

39.10168  =  S  +  0.42543857: 

39.03510  =  S  ■\-  0.09482867: 

39.01884  =  ^  +  0.03414737: 

39.01997  =.  S  -\-  0.02180237: 

39.02074  =  S  -\-  0.00005157: 

39.01214  =  6"  +  0.00194647: 

39.02410  =  S  -\-  0.01903387: 

39.02425  =  6*  +  0.05052017: 

The  normal  equations  formed  from  these  are 

508.18390  =  13.0000006'  +  4.8487027^ 
189.94447  =    4.848702^  +  3.7043947; 

whose  solution  gives 

^  =  39.01568  inches, 
7^=     0.20213  inches, 

as  the  most  probable  values  that  can  be  deduced  from  the  thir- 
teen observations.  Hence  the  length  of  the  seconds'  pendulum 
at  any  latitude,  /,  may  be  written 

s  =  39.01568  +  0.20213  sinV. 

The  values  thus  deduced  for  5  and  T  are  empirical  constants. 
To  find  from  them  the  ellipticity/,  it  is  easily  seen  that 

T 

/  —  5.k  —  — 


§  lO/.  EMPIRICAL    CONSTANTS.  12/ 

and,  as  the  value  of  k  is  known  to  be  ^^  that  of /"is 

/—  0.0086505  —  0.0051807  =  -— -. 

200.2 

If  desired,  the  precision  of  the  constants  5  and  T  may  be 
investigated  by  determining  their  weights  and  probable  errors, 
and  from  these  the  precision  of  the  value  of  f  may  also  be 
inferred. 

107.  When  two  quantities  x  and  y  are  connected  by  the 
relation  y  ^=  Sx -\- T  the  method  of  the  last  article  can,  in 
strictness,  only  be  applied  to  find  the  most  probable  values  of 
5  and  7"  when  the  observed  values  of  x  are  free  from  error.  If 
X  is  liable  to  error  as  well  asj',  the  following  method  may  be 
used.*  First  let  the  value  of  vS"  be  found,  supposing  that  x  is 
without  error,  and  let  this  be  called  S, .  Secondly,  let  the 
value  of  S  be  found  regarding  y  as  without  error,  and  let  this 
be  called  S^.  Let  each  observed  value  of  x  have  the  weight/, 
and  each  observed  value  of  y  have  the  weight  unity.  Then 
the  most  probable  value  of  5  is  found  by  solving  the  quadratic 
equation 

and,  if  n  be  the  number  of  pairs  of  observations,  the  formula 

I 


n 


T  =  -p>-  S.^x 


gives  the  most  probable  value  of  T.     The  following  numerical 
example  will  illustrate  the  method. 

In  order  to  determine  the  most  probable  equation  of  a  cer- 

*  Report  U.  S.  Coast  and  Geodeiic  Survey,  1890,  p.  687. 


y  =  0.5, 

0.8, 

I.O, 

and 

X  =  0.4, 

0.6, 

0.8, 

and 

128  INDEPENDENT  OBSERVATIONS.  VIL 

tain  straight  line  the  abscissas  and  ordinates  of  four  of  its 
points  were  measured  with  equal  precision,  giving 

1.2, 
0.9. 

First,  supposing  that  the  values  of  x  are  without  error,  the 
four  observation  equations  are  written  :  — 

0.5  =  0.46"  +  T, 

0.8  =  0.6^+  T, 

\.o  =  0.86"+  T, 

1.2  =  0.96"  -)-  T. 

And  then,  forming  and  solving  the  normal  equations,  there  is 
found  5j  =  1.339.  Secondly,  supposing  that  the  values  of  y 
are  without  error,  the  equation  of  the  line  must  be  written  in 
the  form 

and  the  observation  equations  are 

o.4  =  o.5C/+r, 
0.6  =  0.8^7+  r, 
0.8  =  1.0  f/+  F, 
0.9  =  1.2^7+  V\ 

from  which  the  normal  equations  are  derived,  and  by  their 
solution  U  =  0.7385,  or  5,  =  1.354. 

These  values  of  5,  and  S,  give  the  quadratic  equation 
5'  —  0.6075—  I  =  O,  whence  S  =  1.348,  and  then  T  is  found 
to  be  —  0.035,  and  accordingly 

y  =  1.348J;  —  0.035 
is  the  most  probable  equation  of  the  line  as  derived  from  the 
four  pairs  of  observations. 

107'.  The  determination  of  the  elements  of  the  orbit  of  a 
comet  or  planet  is  another  instance  of  the  deduction  of  em- 
pirical constants.     Here  the  observed  quantities  are  the  right 


§  I07'.  EMPIRICAL    CONSTANTS.  1 29 

ascension  and  declination  of  the  body  at  various  points  in  its 
orbit.  Through  any  three  of  these  points  a  curve  may  be 
passed,  and  an  orbit  computed  by  the  formulas  of  theoretical 
astronomy.  The  problem,  however,  is  to  determine  the  most 
probable  orbit  by  the  use  of  all  the  observations. 

The  first  step,  after  collecting  and  reducing  the  observations, 
is  to  select  a  few  favorably  situated,  and  from  them  to  compute 
the  approximate  elements  of  an  elliptical  or  parabolic  orbit,  as  the 
case  may  require.  With  these  approximate  elements,  the  places 
of  the  body  are  computed  for  as  many  dates  as  there  are  obser- 
vations, and  the  differences  between  the  computed  and  observed 
places  found.  A  theoretic  differential  formula  is  next  estab- 
lished for  a  difference  in  right  ascension,  and  another  for  a 
difference  in  declination,  in  terms  of  unknown  corrections  to 
the  assumed  elements,  and  of  co-efificients  that  may  be  com- 
puted from  the  observations.  Each  observation  thus  furnishes 
a  difference,  and  each  difference  an  observation  equation,  whose 
unknown  quantities  are  the  corrections  to  the  approximate  ele- 
ments of  the  orbit.  From  the  observation  equations  the  normal 
equations  are  derived  and  solved,  and  the  most  probable  set  of 
corrections  found.  Lastly,  the  application  of  these  corrections 
to  the  approximate  elements  furnishes  the  most  probable  ele 
ments  that  can  be  deduced  from  the  given  observations. 

The  process  thus  briefly  described  is  very  lengthy  in  its 
actual  application.  For  instance,  in  Hall's  determination  of 
the  elements  of  the  orbit  of  the  outer  satellite  of  Mars  *  there 
are  forty-nine  observation  equations,  each  containing  seven 
jnknown  corrections,  and  forty-nine  others,  each  containing 
six.  F"rom  these  the  seven  normal  equations  were  formed,  and 
by  their  solution  the  most  probable  values  found  for  the  correc- 
tions. The  precision  of  the  elements  of  the  orbit  was  also 
deduced  by  computing  the  probable  errors  of  the  corrections. 

*  Hall's  Observations  and  Orbits  of  the  Satellites  of  Mars;  Washington,  1878. 


I30  INDEPENDENT  OBSERVATIONS.  VII. 

Empirical  Formulas. 

io8.  The  case  of  the  last  article  is  that  of  a  rational  formula 
with  empirical  constants.  An  empirical  formula,  on  the  other 
hand,  is  one  assumed  to  represent  certain  observations,  and 
which  is  not  known  to  express  the  law  governing  them.  The 
constants  in  such  formulas  are  also  best  determined  by  the 
application  of  the  Method  of  Least  Squares. 

The  first  step  in  the  establishment  of  an  empirical  formula 
is  to  plot  the  given  observations,  taking  one  observed  quantity 
as  abscissas,  and  the  other  as  ordinates.  Let  y  and  x  be  the 
two  quantities  between  which  an  empirical  formula  is  to  be 
established.  The  plot  shows  to  the  eye  how  j  varies  with  x. 
If  J  is  a  continually  increasing  function  of  x,  or  if  the  curve 
resembles  a  parabola,  the  general  equation 

(40)  y  =  S+  Tx+  C/x^  +Fxi  -\-  etc., 

may  be  written  to  represent  the  relation  between  j  and  x.  This 
equation  applies  to  a  large  class  of  physical  phenomena,  such 
as  relations  between  space  and  velocity,  volume  and  tempera- 
ture, stress  and  strain,  and  other  similar  related  quantities. 
The  letters  S,  T,  U,  etc.,  represent  constants  whose  values  are 
to  be  determined  from  the  observations. 

Another  large  class  of  phenomena  may  be  represented  by 
the  general  equation 

/     \  c  ,    '7'  •    360°       ,    rj.,        360° 

(41)  y  =  S-\-  T%\xi  - — X  +  T  cos  - — x 

m  m 

,-  .     ^60°  ,„        ^60° 

+  t/sin 2x  -\-  U  cos 2x  +  etc., 

m  m 

in  which,  as  x  increases,  y  passes  through  repeating  cycles.  As 
such  may  be  mentioned  the  variation  of  temperature  through- 
out the  year,  the  changes  of  the  barometer,  the  ebb  and  flow 
of  the  tides,  the  distribution  of  heat  on  the  surface  of  the  earth 
depending  on  latitude,  and,  in  fact,  all  phenomena  which  repeat 


§  109-  EMPIRICAL   FORMULAS.  I31 

themselves  like  the  oscillations  of  a  pendulum.  The  letters 
5,  T,  U,  etc.,  represent  constants  which  are  to  be  found  from  the 
observations  ;  while  m  is  the  number  of  equal  parts  into  which 
the  whole  cycle  is  divided,  and  must  be  taken  in  terms  of  the 
same  unit  as  x.  If  the  several  cycles  are  similar  and  regular,  only 
the  first  three  terms  are  required  to  represent  the  variation. 

Other  general  empirical  formulas  than  (40)  and  (41)  are  also 
employed  in  discussing  physical  phenomena.  Exactly  what 
formula  will  apply  to  a  given  set  of  observations,  so  as  to  agree 
well  with  them,  and  at  the  same  time  be  of  use  in  other  similar 
cases,  can  only  be  determined  by  trial.  The  investigator  must, 
from  his  knowledge  of  physical  laws,  assume  such  an  expression 
as  seems  most  plausible,  and  then  deduce  the  most  probable 
values  of  the  constants.  The  comparison  of  the  observed  and 
calculated  results  furnishes  the  residuals,  from  which,  if  desired, 
the  probable  errors  may  be  deduced.  When  several  empirical 
formulas  have  been  determined  for  the  same  observations,  that 
one  is  the  best  which  furnishes  the  smallest  value  for  the  sum 
of  the  squares  of  the  residuals. 

109.  Consider  as  a  first  practical  example  the  deduction  of  the 
equation  of  the  vertical  velocity  curve  for  the  observations  given 
on  p.  244  of  the  second  edition  of  the  "  Report  on  the  Physics 
and  Hydraulics  of  the  Mississippi  River,"  by  Humphreys  and 
Abbot.  The  grand  means  of  the  measurements  give  the  following 
results  for  the  velocities  at  different  depths  below  the  surface  : 

At       surface,  3.1950  feet  per  second. 

At  0.1  depth,  3.2299  feet  per  second. 

At  0.2  depth,  3.2532  feet  per  second. 

At  0.3  depth,  3.261 1  feet  per  second. 

At  0.4  depth,  3.2516  feet  per  second. 

At  0.5  depth,  3.2282  feet  per  second. 

At  0.6  depth,  3.1807  feet  per  second. 

At  0.7  depth,  3.1266  feet  per  second. 

At  0.8  depth,  3.0594  feet  per  second. 

At  0.9  depth,  2.9759  feet  per  second. 


132 


INDEPENDENT  OBSERVATIONS. 


VI T 


2.9 

3.0 

3 1               3.2               3.3 

0.1 

®N 

\ 

0^ 

\ 

0.S 

\ 

0.4 

I 

0.5 

/ 

0.6 

./ 

r 

0.7 

X 

0.8 

rr^ 

/'-^ 

0.9 

y^ 

^ 

These  observations  may  be  plotted  by  dividing  a  vertical  line 
representing  the  depth  of  the  river  into  ten  equal  parts,  through 

the  points  of  division  drawing 
horizontal  lines,  and  laying 
off  upon  these  the  observed 
velocities.  On  the  annexed 
figure  the  points  enclosed 
within  small  circles  represent 
the  observations.  Each  hori- 
zontal division  of  the  diagram 
is  o.  I  feet  per  second,  and 
each  vertical  division  is  one- 
tenth  of  the  depth. 

Let  y  be  the  velocity  at 
any  point  whose  depth  below 
the  surface  is  x,  the  total 
depth  of  the  river  being  unity,  and  assume  that  three  terms  of 
formula  (40)  will  give  the  relation  between  y  and  x,  or  that 

y  =  S-^  Tx^-  Ux\ 

This  is  equivalent  to  assuming  that  the  curve  of  vertical  veloci- 
ties is  a  parabola,  with  its  axis  horizontal. 

The  observations  furnish  the  values  of  y  for  ten  values  of  x  ; 
and  thus,  for  determining  5,  T,  and  U,  there  are  the  following 
ten  observation  equations  :  — 

3.1950  =  S  -\-  o.oT  +  o.ooU. 

3.2299  =  S  ■\-  o.iT  -\-  o.oi  U. 

3.2532   =   6"  +  0.27"+  O.Oj^U. 

3.261 1  =  'S  +  0.3  7"+  o.o^U. 
3.2516  =  S  -\-  0.47"+  0.16  U. 
3.2282  =  .^S"  +  0.57"+  0.25  6^". 
3.1807  =  S  +  0.6T+  o.^GU. 
3.1266  =  S  +  o.'jT -\-  0.496^. 
3.0594  —  ^  -f-  o.d>T -\-  0.64(7. 
2.9759  =  S  -j-  o.gT -j-  o.SiU. 


§  1 09.  EMPIRICAL   FORMULAS.  1 33 

From  these  the  following  three  normal  equations  are  found  : 

31.761600  =  10.006"+  4.5007"+  2. 85006^. 

14.089570  =    4.506"+  2.8507"+  2.02506^. 

8.828813=    2.856'+  2.0257"+ 1.53336: 

And  their  solution  gives 

S=  +  3-I95I3.         ^=  +  0.44253,         ^=  —  0.7653. 

Accordingly,  the  empirical  formula  of  vertical  velocities  is 

y  =  3-19513  +  044253-^"  -  0-7653^", 

where  y  is  the  velocity  in  feet  per  second  at  any  decimal  depth 
X.  The  curve  corresponding  to  this  formula  is  drawn  on  the 
above  diagram. 

The  following  is  a  comparison    of   the  observed  velocities 
with  those  computed  from  this  empirical  formula : 


X. 

Observed^. 

Computed  7. 

V. 

v". 

0.0 

3-1950 

3-I951 

—  0.000 1 

0.000000 

O.I 

3.2299 

3-2317 

—  0.0018 

3 

0.2 

3-2532 

3-2530 

+  0.0002 

0 

0-3 

3.261I 

3-2590 

+  0.0021 

4 

0.4 

3-2516 

3-2497 

+  0.0019 

4 

0-5 

3.2282 

3-2251 

+  0.0031 

10 

0.6 

3.1S07 

3-1851 

—  0.0044 

19 

0.7 

3.1266 

3.1299 

—  0.0033 

II 

0.8 

3-0594 

3-0594 

0.0000 

0 

0.9 

2-9759 

2-9735 

+  0.0024 

6 

I.O 

2.8724 

134 


INDEPENDENT  OBSERVATIONS. 


VI 


The  sum  of  the  squares  of  the  residuals  is  here  0.000057,  and 
hence 


^  /0.000057 
=  0.6745V ^  =  0.0019 

T         10  —    ^ 


is  the  probable  value  of  a  residual,  or  the  probable  difference 
between  an  observed  and  computed  velocity.  The  agreement 
between  the  parabola  and  the  observed  points  is  very  close.* 

no.  As  a  second  example,  consider  the  deduction  of  a 
formula  to  express  the  magnetic  dechnation  at  Hartford, 
Conn.,  for  which  place  the  following  observations  are  given 
on  p.  225  of  the  United  States  Coast  and  Geodetic  Survey 
Report  for  1882  : 


Date. 

Dec 

ination. 

1786 

5° 

25'   w. 

1810 

4 

46 

1824 

5 

45 

1828-29 

6 

03 

27  July,    1859 

7 

17 

16  Aug.,  1S67 

7 

49-3 

25  July,  1879 

8 

340 

From  numerous  records  at  various  places,  it  is  known  that  the 
declination  oscillates  slowly  to  and  fro,  passing  through  a  cycle 
in  a  period  varying,  at  different  places,  from  two  hundred  and 
fifty  to  four  hundred  years. 


The  variation  in   New  England 


*  See  further,  concerning  this  curve,  in  Journal  Franklin  Institute,  1877,  vol.  civ, 
p.  233;  also  Van  Nostrand's  Magazine,  1877,  vol.  xvii,  p.  443,  and  1878,  vol.  xviii, 
p.  I.  The  reasoning  of  Hagen  concerning  the  probable  errors  on  p.  447  of  the 
second  article  is  thought  to  be  incorrect. 


EMPIRICAL   FORMULAS. 


135 


may  be  roughly  represented  by  the  annexed  figure,  where  the 
ordinates  to  the  curve  show  the  relative  values  of  the  declina- 
tion at  the  respective  years.  Formula  (41)  is  hence  applicable 
to  the  discussion  of  the  above  observations. 


1000 


\'i\M 


1»U(I 


lyou 


Let  y  be  the  magnetic  declination  at  the  time  x,  and  assume 
the  empirical  relation 

360°            ,       360° 
y  =  S  -\-  T  %\n  ~ X  +  T  cos x. 


7)1 


m 


Here  there  are  four  constants,  S,  T,  T',  and  vi,  to  be  found  by 
the  Method  of  Least  Squares  from  the  given  observations. 
The  only  practical  way  of  procedure  is  to  assume  a  plausible  value 
of  in,  and  then  to  state  the  observation  equations  and  normal 
equations,  from  which  values  of  S,  T,  and  T'  may  be  deduced. 
Again  :  assume  another  value  of  ni,  and  repeat  the  computation, 
thus  finding  other  values  for  S,  T,  and  T'.  If  necessary,  the 
computation  is  to  be  repeated  for  several  values  of  ;;/;  and  for 
each  formula  thus  deduced  the  residuals,  or  differences  between 
the  observed  and  computed  values  of  j,  are  to  be  found.  Then 
that  value  of  in  and  that  formula  is  the  best  which  makes  the 
sum  of  the  squares  of  the  residuals  a  minimum. 

Take  for  ;;/  the  value  288  years  ;  then is   1,25,  and  the 


formula  is 


;;/ 


6"  +  7"sin  1.25J1:  +  7"  cos  1.25.x  =  y. 


Let  X  be  the  number  of  years  counted  from  the  epoch,  Jan.  i, 


136 


INDEPENDENT  OB  SEE  V A  TIONS. 


vu 


1850,  and  let  all  angles  be  expressed  in  degrees  and  decimals; 
then,  for  the  first  observation, 

X  =  1786.5  —  1850.0  =  —63.5  years, 

i.25.r  =  —79.4  degrees, 

sin  1.25X  =  —0.983, 

cos  1.25JC  =  +0.184, 

y  =  5.42  degrees, 

and  hence  the  first  observation  equation  is 

^  -  0.9832"  +  0.1847^'  =  5.42. 

In  like  manner  the  following  tabulation  is  made 


No. 

Date. 

X. 

1.25X. 

.Sin  1.25X. 

Cos  1.25X. 

y- 

I 

1786.5 

-63.5 

-79°-4 

-0.983 

+  C.184 

+  5°42 

2 

1810.5 

-39-5 

-49.4 

-0-759 

+  0.651 

+  4-77 

3 

1824.5 

-25-5 

-31-9 

-0.528 

+  0.849 

+  5-75 

4 

1829.0 

—  21.0 

-26.25 

-0.442 

+  0.897 

+  6.05 

5 

1859.6 

+   9-6 

+  12.0 

+0.208 

+  0.978 

+  7.29 

6 

1867.6 

+  17.6 

+  22.0 

+  0-375 

+  0.927 

+  7-82 

7 

1879.6 

+  29.6 

+  37-0 

+  0.602 

+  0.799 

+  8.57 

From  the  last  three  columns  the  seven  observation  equations 
are  written ;  and  from  these  the  three  normal  equations  are 
easily  formed,  either  by  the  rule  of  Art.  48,  or  by  the  help  of 
the  algorithm  of  formulas  (10)  and  (11).     They  are 

+  7.005-  1.53^+  5-28r-  45-67  =  o, 
-1.53S+  2.s6T-o,siT'+    5-03  =  0, 

+5.285-  0.51  r+  +53^'-  35-64  =  o, 

and  their  solution  gives 

S=  +8°.o6,     T=  +2°.6o,     7"=  -i°.29. 


§  no. 


EMPIRICAL   FORMULAS. 


m 


Hence  the  empirical  formula  is 

y  =  8°. 06  +  2°.6osin  1,25a:  —  1°. 29  cos  1.25X. 

This  may  also  be  written 

y  —  +8°.o6  +  2°.90sin(i°.25.r  —  26''.4), 

which  is  a  more  convenient  form  for  discussion.* 

The  following  is  a  comparison  of  the  observed  declinations 
with  those  computed  from  this  formula  : 


Date. 

X. 

Observed  y. 

Gomputed  J. 

V. 

1786.5 

-63-5 

+  5°-42 

5^28 

+0.14 

1810.5 

-39-5 

4-77 

5-25 

—0.48 

1824.5 

-25-5 

5-75 

5.60 

+0.15 

1829.0 

—  21.0 

6.05 

5-76 

+0.29 

1859.6 

+   9-6 

7.29 

7-34 

-0.05 

1867.6 

+  17-6 

7.82 

7.84 

—  0.02 

1879.6 

+  29.6 

8.57 

8.59 

—  0.02 

The  sum  of  the  squares  of  these  residuals  is  0.36,  and  hence, 

by  (32), 


'^  7  -  3 


which  gives  the  probable  error  of  a  single  computed  value  if 
the  observations  be  regarded  as  exact,  or  the  probable  error 
of  an  observation  if  the  law  expressed  in  the  empirical  formula 
be  regarded  as  exact. 

*  See  the  numerous  valuable  papers  by  Schott,  in  the  Reports  of  the  United 
States  Coast  and  Geodetic  Survey,  the  latest  of  which  is  in  the  Report  for  1882, 
pp.  211-276.  The  above  formula  for  the  declination  is  the  one  there  adopted,  as 
givmg  the  best  value  of  the  period  m. 


138 


INDEPENDENT  OBSERVATIONS. 


VII. 


III.  Lastly,  consider  the  deduction  of  a  formula  to  represent 
certain  experiments,  made  by  Darcy  and  Bazin,  on  the  flow  of 
water  in  a  rectangular  wooden  trough  lined  with  cement.  The 
width  of  the  trough  was  1.812  meters,  and  its  slope  0.0049. 
Water  was  allowed  to  run  through  it  with  varying  depths  ;  and 
for  each  depth  the  mean  velocity  was  measured,  and  the  hydrau- 
lic radius  of  the  water-section  computed  by  dividing  the  wetted 
perimeter  into  the  area  of  the  section.  The  following  are 
the  results,  the  hydraulic  radius /^  being  given  in  meters,  and  the 
mean  velocity  m  in  meters  per  second  : 


No. 

h. 

VI. 

I 

0.1 144 

1-731 

2 

.1312 

1-853 

3 

•  1445 

1.9S4 

4 

•1579 

2.081 

5 

.1  701 

2.1  71 

6 

.1813 

2.258 

7 

.1925 

2.326 

8 

.2026 

2-397 

9 

.2123 

2.460 

Assume  the  expression 


m 


shf. 


and  let  it  be  required  to  find  from  the  above  experiments  the 
most  probable  values  of  s  and  t.  First  reduce  the  expression 
to  a  linear  form  by  writing  it  thus  : 

\ogtn  =  log^  +  t\ogh. 

Each  observation  furnishes  an  observation  equation  containing 
log  s  and  t.     For  example,  the  first  is 

0.2383  —  \ogs  —  0.9416/. 


§112. 


PROBLEMS. 


139 


The  twelve  observation  equations  furnish  the  two  normal  equa- 
tions, and  their  solution  gives 

t  =  0.572,     log^  =  0.7767,     .-.  s  =  5.98. 

Therefore  the  empirical  formula 

m  =  5.98/i°-572 

is  the  best  of  the  assumed  form  that  can  be  derived  from  the 
nine  experiments. 

112.   Problems. 

1.  The  following  levels  were  taken  to  determine  the  elevations  of  five 

points,  T,  U,  IF,  X,  and  V,  above  the  datum  O  : 

T  above  O  =  1 15.52. 
U  above  7"=  60.12. 
U  above  O  —  177.04. 
^  above  7"=  234.12. 
W  above  ^/  =  171 .00. 

What  are  the  adjusted  elevations? 

Ans.   T—  115. 61,     U=  176.95,     etc. 

2.  Four  angles  are  observed  at  a  station,  and  also  their  sum.  The 
observed  sum  differs  from  the  sum  of  the  four  observed  parts  by  the 
discrepancy  d.     What  are  the  adjusted  values? 

3.  Adjust  the  following  angles,  taken  at  the  station  Moodus,  and  find 
the  probable  errors  of  the  adjusted  values. 


X  above  W  —  632.25. 
Jf  above  Y  =  211.01. 

Y  above  U  —  596.12. 

Y  above  ^F  =  427.18. 


No. 

Between  Stations. 

Observed  Angle. 

Weight. 

I 

Big  Rock  and  Small  Rock 

99°  42'  i5"-6i 

137 

2 

Small  Rock  and  Tokus 

^33    39    05.07 

22 

3 

Small  Rock  and  Buzzard 

40    12    52.43 

57 

4 

Buzzard  and  Tokus 

93    26    13.14 

50 

5 

Tokus  and  Big  Rock 

1 26    38    40.69 

20 

Ans.  99°  42' i5".46,  etc. 


140  INDEPENDENT  OBSERVATIONS.  VII. 

4.  The  following  observations  of  the  temperature  at  different  depths 
were  taken  at  the  boring  of  the  deep  artesian  well  at  Crenelle  in  France, 
the  mean  yearly  temperature  at  the  surface  being  10°. 60  centigTade  : 

1.  Temperature  at  a  depth  of    28  meters  =  11. 71  degrees. 

2.  Temperature  at  a  depth  of    66  meters  =  12.90  degrees. 

3.  Temperature  at  a  depth  of  173  meters  =  16.40  degrees. 
4    Temperature  at  a  depth  of  248  meters  =  20.00  degrees. 

5.  Temperature  at  a  depth  of  298  meters  =  22.20  degrees. 

6.  Temperature  at  a  depth  of  400  meters  =  23.75  degrees. 

7.  Temperature  at  a  depth  of  505  meters  =  26.45  degrees. 

8.  Temperature  at  a  depth  of  548  meters  =  27.70  degrees. 

Deduce  from  these  observations  the  empirical  formula 
/  =  10". 6  -|-  0.041 5x  —  0.0000193x2, 

where  t  is  the  temperature  at  a  depth  of  ,v  meters. 

5.  Gordon's  formula  for  the  ultimate   strength  of  columns  may  be 

written 

_       S 

'  ~  1  +  Tj'' 

in  which  c  is  the  crushing-load  per  unit  of  area  of  cross-section,  /  the 
ratio  of  the  length  of  the  column  to  its  least  diameter,  and  S  and  T  are 
constants  to  be  found  by  experiment.  Determine  the  best  values  of 
these  constants  for  the  following  four  experiments  on  wrought-iron 
Phoenix  columns  : 

c  =  34650,     35000,     36580,     37030. 
J-        42,  Zo^  24,  19.5. 

6.  From  several  census  records  of  the  papulation  of  the  United 
States  deduce  an  empirical  formula  shovving  the  population  for  any 
year. 


§113.  METHOD   OF  PROCEDURE,  I4I 


CHAPTER  VIII. 

CONDITIONED   OBSERVATIONS. 

113.  The  general  method  of  adjusting  conditioned  observa- 
tions has  been  deduced  in  Arts.  56,  57,  and  that  of  investigating 
the  precision  in  Arts.  TJ,  78.     The  following  is  the  process  : 

1st,  Having  given  n  observations  upon  q  quantities  subject 
to  ;/  rigorous  conditions,  the  first  step  is  to  represent  the  quan- 
tities by  symbol-s,  and  state  ;/  observation  equations  and  ;/  con- 
ditional equations.  Generally  it  will  be  found  most  convenient 
to  take  the  unknown  quantities  as  representing  corrections  to 
assumed  approximate  values,  and  to  state  the  observation  and 
conditional  equations  in  terms  of  these  corrections. 

2d,  From  the  ;/  conditional  equations  find  the  values  of  ;/ 
unknown  quantities  in  terms  of  the  remaining  q  —  n'  quanti- 
ties, and  substitute  these  values  in  the  n  observation  equations, 
each  of  which  then  represents  an  independent  observation. 

3d,  Adjust  these  n  observation  equations  by  the  method  of 
Chap.  VII,  and  find  the  most  probable  values  of  the  q  —  n' 
quantities.  Then,  by  substitution  in  the  conditional  equations, 
the  most  probable  values  of  the  remaining  n'  quantities  are 
known. 

4th,  Insert  the  adjusted  values  in  the  11  observation  equa- 
tions, and  find  the  residuals,  and  then,  from  {33),  the  probable 
error  of  an  observation  of  the  weight  unity.  If  desired,  the 
weights  of  the  adjusted  values  may  be  found  by  Art.  75,  and 
their  probable  errors  by  (31). 


142  CONDITIONED    OBSERVATIONS.  VIII. 

114.  The  special  method  of  correlatives,  which  is  particu- 
larly valuable  in  the  adjustment  of  geodetic  triangulations,  has 
been  explained  in  Art.  58.  In  order  to  apply  it,  the  local 
adjustments  should  first  be  made;  so  that  for  each  quantity, 
s',,  XT,  ...  2-^,  a  value,  J/„  M-,  .  .  .  Mg,  called  the  observed  value, 
is  known.  The  numbers  q  and  ;/  are  hence  equal.  The  fol- 
lowing are  the  steps  of  the  practical  application  : 

I  St,  For  the  rigorous  conditions  write  ;/  conditional  equa- 
tions, as  in  (14).  Substitute  in  these  the  observed  values,  M„ 
M2  .  .  .  Mg,  for  the  quantities  :;„  s^  .  .  .  Zg\  and  let  d„  ^/,  .  .  .  dg 
be  the  differences  or  discrepancies  that  arise. 

2d,  Assume  ;/  new  unknown  quantities,  or  correlatives, 
K^,  K2  .  .  .  K„',  and  write  the  normal  equations  (16).  Solve 
these  normal  equations,  and  thus  find  the  values  of  the 
correlatives. 

3d,  From  (15)  find  the  corrections  v„  v^  .  .  .  Vg,  which,  when 
applied  to  the  observed  values  AT,,  M^  •  •  .  Mg,  give  the  most 
probable  adjusted  values. 

4th,  Compute  the  sum  2/■^;^  and  from  (34)  find  the  proba- 
ble error  of  an  observation  of  the  weight  unity.  The  probable 
error  of  any  observed  M  is  then  easily  found  from  (31),  and 
that  of  the  corresponding  adjusted  value  is  somewhat  smaller, 
since  the  weights  are  increased  by  the  adjustment. 


Angles  of  a   Triangle. 

115.  When  the  three  observed  angles  of  a  plane  triangle  are 
of  equal  weight,  it  is  easy  to  show  that  the  correction  to  be 
applied  to  each  is  one-third  of  the  discrepancy  between  their 
sum  and  180°.  The  following  is  the  proof  by  the  method  of 
correlatives  : 

1st,  Let  M^,  M^,  and  M^  be  the  observed  values,  and  z„  r„ 


§Il6.  ANGLES   OF  A    TRIANGLE.  1 43 

and  z^  the  required  most  probable  values.      The   conditional 
equation  is 

Si  +  2^2  +  ^3  —  i8o°  =  o- 

Substitute  in  this  the  observed  values,  and  it  does  not  reduce 
to  zero,  but  leaves  a  small  discrepancy  d ;  thus 

li\  +  M^  +  M^  -  i8o°  =  d. 

By  comparison  with  (14)  it  is  seen  that  a,  =  a^  =  aj  =  -f-  I- 

2d,  Take  K  as  the  single  correlative.     The  weights  are  all 
equal,  or/  =  i.     From  (16)  the  single  normal  equation  is 

[affJ.A'  4-^=0,    or    3^^  +  ^  =  o. 


from  which  K  ^ 

-¥■ 

3d, 

From  (15) 

the  three  corrections 

now  are 

V,  = 

d 
3 

d 
3 

^'3=    - 

> 

3 

and,  accordingly,  the  most  probable  values  of  the  three  angles 
are 


d 

d 

,..       d 

z,  =  M,  - 

J 

^2  =   ^^r. 

J 

h  =  ^^3  -  -  • 

3 

3 

3 

4th,  The  sum  of  the  squares  of  the  residuals  is  — ,  and  hence 

by  (34)  the  probable  error  of  a  single  c  bserved  angle  is  0.39^. 
By  working  the  problem  according  to  the  general  method  of 
Art.  113,  it  may  be  shown  (as  in  Art.  103)  that  the  pr3bable 
error  of  an  adjusted  angle  is  0.32^/. 

116.    When  the  three  observed  angles  of  a  plane  triangle  are 
of  unequal  weights,  it  is  easy  to  show  that  the  corrections  to  be 


144 


COND I TIONED    OBSER  VA  TIONS.  VIII. 


applied  are  inversely  as  the  weights.     For  instance,  take  the 
following  numerical  case  : 

M,  =  36°  25'  47",  with  weight  4 
M-i  =  90  36  28,  with  weiglit  2 
M^=     52     57       57,        with  weight  3 

Sum  —  180°  00'      12" 

1st,  Take  z,,  z,,  and  z^  as  the  most  probable  values;  then, 
as  before,  the  conditional  equation  is 

2i  4-  ^2  +  23  —  1 80°  =  o. 

The  discrepancy  is   12".     To  compare  with  (14),  (15),  and  (16), 
a,  =a^  =  a3  =  4-  I,    /.  =  4,    A  =  2,     and    A  =  3- 

2d,  Only  one   correlative    is   necessary ;  and  from   (16)   the 
single  normal  equation  is 

and  hence  A'  =  —  ^  =  —  1 1.08. 

3d,  From  (15)  the  corrections  now  are 

z;,  =  —  =  -  2".77,         v^=-s"-SA,         v^=''f'^9» 
4 

and  the  adjusted  angles  are 

2.  =    36°  25'  44"-23 

Z2  =    90  36  22.46 

h=    52  57  53-31 

Sum  =  180°  00'  oo".oo 

4th,  The  residuals  are  the  three  corrections  v„  v^,  and  v^ 
and  the  sum  of  their  weighted  squares  is  '^pv^  =  132.92,  from 
which,  by  (34),  r  =  f.^y  for  the  probable  error  of  an  observa- 


§Il8.  ANGLES  AT  A   STATION:  I45 

tion  of  the  weight  unity.     By  (31)  the  probable  errors  of  the 
observed  values  are  found  to  be 

^i  =  3"-S9.  ^^  =  5"-5o.  ^3  =  4"-49, 

and  the  probable  errors  of  the  adjusted  values  are  somewhat 
less  than  these. 

The  adjustment  of  the  angles  of  a  spherical  triangle  differs 
from  that  of  a  plane  triangle  only  in  the  introductic^n  of  the 
spherical  excess  into  the  conditional  equation  ;  thus  s  -\-  f  -\-  u 
=  180°  +  spherical  excess. 

Auirles  at  a  Station. 


'<b 


117.  When  }i  angles,  and  also  their  sum,  are  observed  at  a 
station,  and  the  weights  are  all  equal,  it  is  easy  to  show,  as  in 
Art.   103,  that  the  correction  to  be  applied  to  each  observed 

angle  is  th  of  the  discrepancy  between  the  observed  sum 

and  the  sum  of  the  observed  single  angles. 

When  71  angles,  which  close  the  horizon,  are  observed  at  a 
station,  and  the  weights  are  equal,  it  is  easy  to  show,  as  in 
Art.   115,  that  the  correction  to  be  applied  to  each  observed 

angle  is  -th  of  the  discrepancy  between  360°  aad  the  sum  of 
the  observed  angles. 

When  angles  at  a  station  close  the  horizon,  or  are  observed 
by  sums  or  differences,  the  adjustment  may  be  effected,  either 
for  equal  or  unequal  weights,  by  the  method  of  Art.  J  i  3,  or  by 
that  of  Art.  114.  The  former  will  always  reduce  tc  the  method 
of  independent  observations,  as  exemplified  in  Arts.  103-105. 

118.  As  an  example  of  the  application  of  the  method  of  cor- 
relatives, consider  the  observations  of  Art.  104.  Represent  the 
most  probable  values  of  the  seven  angles  by  ^„  2^  .  .  .  Zj. 


146  CONDITIONED    OBSERVATIONS.  VIII. 

From  Fig.  7  the  following  conditions  are  seen  : 

2,  —  S2  +  23  =  o, 

24  —  Sj  +  26  +  Z7  =0, 

Z,   +  23   4-  2^  +  26  +  2;  —  360°  ^  O. 

By  substituting  in  these  the  observed  values,  the  following  dis- 
crepancies are  found:  — 

^,  =  —  0.210,         ^/j  =  —  0.648,        ^3  =  —  0.420. 

Take  A^,,  K^,  and  K^  as  the  correlatives  to  be  determined. 
By  comparison  with  (14),  it  is  seen  that 

a,  =  +  I,      a,  =   —  I,      ttj  =  +  I,      a^  =  ttj  =  06  =  a^  =  O, 

A  =  /?.  =  ^3  =  o,     /3,==/?6  =  /3,=  +  i,     ^5=  -I, 
y.  =  73  =  74  =  76  =  7?  ==  +  1'     72  =  75  =  o- 

All  weights  are  unity.     The  three  normal  equations  then  are, 
from  (16), 

T^K^  +  2K^  —  0.210  =  o, 

+  a^K.  +  Z^i  ~  0.648  =  o, 

2A',  -|-  TyK^  +  5A'3  —  0.420  =  o, 

and  their  solution  gives 

A",  =  +  0.167,        A',  =  +  0.271,        K^——  0.145. 

From  (15)  the  corrections  now  are 

Z'l  =  +  A",  +  A'3  =  +  o".02  2, 

2^2  =  —  A',  =  —    0.167, 

2/3=+  A',  +  A'3  =  +    0.022, 
2^4  =+  ATj  +  A'3  =  +    0.126, 

2^5  =   —  A'2  =   —     0.271, 

Vb  =  +  A',  +  A'3  =  +    0.126, 
Vj  =  +  A',  -\-  A\  =  -f-    0.126, 


§U9- 


ANGLES  OF  A    QUADRILATERAL. 


147 


and  if  these  be  applied  to  the  observed  values  M^,  M^  .  .  .  M^, 
the  adjusted  values  are  found  the  same,  within  Oiie  or  two  thou- 
sandths of  a  second,  as  in  Art.  104,  the  slight  difference  being 
due  to  the  neglect  of  the  fourth  decimal  places. 


Angles  of  a  Quadrilateral. 

119.    In  a  quadrilateral    JVXVZ,  the    two    single   angles  at 
each  corner  are  equally  well  measured.     It  is  required  to  ad- 
just them,  so  that  the  sum  of  the  three  angles  in  each  triangle 
shall  equal   180°,  and  the  sum  of 
the  four  angles  of  the  quadrilater- 
al shall  equal  360°. 

Let  the  measured  angles  at  the 
corner  W  be  denoted  by  JF,  and 
IV2,  and  similarly  for  each  of  the 
other  corners,  as  shown  in  Fig.  10. 
Let  2v^  and  zv^  be  corrections  to 
be  applied  to  IV,  and  IV-,  in  order 
to  give  the  most  probable  values, 
W,  +  ZL\  and  JT^  -|-  zu-,. 

In  order  to  avoid  writing  identical  equations,  select  any 
corner,  as  IV,  and  take  the  three  triangles,  WXZ,  ZIVY,  and 
XYW  which  meet  at  that  point,  as  the  three  triangles  for  cor- 
rection. Evidently,  if  the  angles  of  these  triangles  add  up  to 
180°,  those  of  the  fourth  triangle  will  also.  The  three  con- 
ditional equations  now  are 

Z^i  +  7^2  +  X,  -f  S,  +  rt'i  =  o, 
W2+  X,  +  X^  +}\  +  </3  =  o, 
a/.  +  ^2     +  2.    +  2,  +  a'j  =  O, 


Fig.lO. 


vn  which  d^,  d^,  and  d^  denote  the  differences  or  discrepancies 


148  CONDITIONED    OBSERVATIONS.  VIII. 

between  the  sum  of  the  measured  angles  of  the  triangles  and 
the  theoretic  sum  180°;  thus,  for  example, 

W,  +  ]V\  +  X,  +  Z,  -  iSo°  =  d,. 

From  the  three  conditional  equations  the  values  of  the  eight 
corrections  are  to  be  found,  either  by  the  method  of  Art.  113 
or  by  that  of  Art.  114.  The  latter  will  be  the  shorter.  As- 
sume, then,  three  correlatives,  K^,  K^,  and  K^,  and  for  each 
correction  write  a  correlative  equation,  thus 

+  K,  +  K^  =  a/„ 

+  A',  +  A',  =  W2, 

+  AT,  +  K:,  =  Xi, 

•{■  K^  +  A'3  =  2,, 

+  ATz  =  ^2, 

/  -V  K2  =  j'„ 

+  A'3   =   Si, 

the  co-efficients  of  K^  being  the  co-efficients  of  the  corre- 
sponding unknown  quantities  in  the  first  conditional  equation, 
and  so  on.  From  these  equations  the  three  normal  equa- 
tions are 

4A'  -f  2K.  +  2A'3  -f  d,  =  o, 

2K,  +  4A',  +  d^  =  o, 

2Ar,  +  4A'3  +  d^  =  o, 

whose  solution  gives  the  values  of  K„  K^,  and  A'3 ;  and,  insert- 
ing these  in  the  correlative  equations,  the  following  values  of 
the  corrections  are  found  : 

Wj  =  Z2  =  ^{—  2d,  -^  d.  —  dj), 
W2  =x,  =  ^{—  2d,  —  d^  +  ^3), 
^2=}\  =  i{  2d,  —  id2—  4), 
J2  =  2i  =  i(       2^1  —    4  —  34), 


§  I2C;  ANGLES  OF  A    QUADRILATERAL.  1 49 

and  the  addition  of  these  to  the  observed  angles  gives  the 
adjusted  values. 

For  example,  let  the  following  angles  be  given  : 

^,  =  41°  58'  47",  J'.  =  49°   17'  30", 

W^^dA,  oS  34,  K  =  53     53  51. 

Xi  =  36  34  i5j  ^^  =  46     49  16, 

^2  =  29  59  51,  ^2  =  37     iS  18. 

Here  the  discrepancies  are 

d,  =  IK  4-  m  +  X,  +  Z  -  180°  =  -  6", 
^,  =.  J II  +  X,  +  a;  4-  K,  -  180  =4-10, 
4  =  /^.  +  i;  +  Z.  4-  Z,  -  180   =  +12. 

I'hen,  by  the  above  formulas,  the  corrections  are 

7a,  =  z,=  +  i".25,  W2  =  x,=  +  i".75, 

^2=}\=-       6.75,  J3     =    2,    =     -        7.25, 

SO  that  the  adjusted  values  are 

JV,-\-w,  =  41°  58'  48".25,  y.  +  J.  =  49"  17'  23".2S, 

^2  +  0^2  =  64  08     35.75,  F3+J,  =  53  53     43.75, 

JT,  4-  .r.  =  36  34      16.75,  Z,  4-  s,  =  46  49     08.75, 

^2  +  ^2  =  29  59     44.25,  Z^  +  S2  =  37  18     19.25. 

These  angles  now  fulfil  all  the  geometrical  conditions  required 
in  the  statement  of  the  problem,  and  are,  furthermore,  the 
most  probable  angles. 

120.  If  the  large  angles  at  the  corners  are  measured  as  well 
as  the  single  angles,  the  most  convenient  method  of  procedure 
is,  first  to  make  the  station  adjustment  at  each  corner,  and  then, 
with  the  eight  single  angles,  to  make  a  further  adjustment,  as  in 
the  last  article.     The  following  is  an  example  illustrating  the 


150  CONDITIONED   OBSERVATIONS.  VI 1 1. 

steps  of  the  process  for  the  case  of  unequal  weights.     Let  the 
twelve  measured  angles  be 

W  =  106°  07'  27",  weight  3,  K  =  103°  11'  15",  weight  2, 

/F,  =     41   58  47,    weight  3,  y,  =     49   17  30,    weight  2, 

W^=     64  oS  34,   weight  3,  K2  =     53  53  51,    weight  2, 

X  ■=     66  34  03,    weight  i,  Z  =     84  07  30,    weight  4, 

X,  =     36  34  21,   weight  I,  ^,  =     46  49  16,    weight  i, 

^2  =     29  59  45,    weight  I,  Z2=     371818,   weight  2. 

First,  by  Art.  117,  make  the  station  adjustment  at  each  corner, 
and  obtain  the  followino:  results  : 


'O 


W,  =  41°  58'49".o,  weight  f,  Y,  =  49°  i7'28".o,  weight  3, 

W^  =    64  08    36.0,  weight  f,  V,  =    53    53    49.0,  weight    3, 

^i  =    36  34    20.0,  weight  |,  Z,  =    46    49    13.7,  weight   |, 

-^2=    29  59    44.0,  weight  3,  Z,  =    37    18    16.9,  weight -1^. 

Next  let  w,,  w^,  etc.,  be  corrections  to  these  values  in  order  to 
satisfy  the  geometrical  tequirements  of  the  figure.  Then,  as 
in  the  preceding  article,  the  three  conditional  equations  are 

u>,  +  uf^  +  x\  +  52  +  i"-9  =  o> 
71^2+  x,  +  X2  +  }\  +  8.0  =  o, 
w,  +  y^  +  z,  +  S2+    8.6  =  o. 

From  (15)  the  eight  correlative  equations  are 


Wi  = 

|(^.                +^3), 

W2  = 

|(A-.  +  A'2           ), 

X,    = 

|(A',  +  A'2          ), 

X2  = 

f  (       +  A^.          ), 

J.  = 

K      +^2         )» 

y2  = 

i(                +  A'3), 

2.  = 

f(                +^'3), 

Zz     = 

^5,  (A-.            +A'3). 

§  121  ANGLES  OF  A    QUADRILATERAL.  151 

From  (16)  the  three  normal  equations  now  are 

(f  +  -I  +  f  +  A) A'.  +  (f  +  !) a;  4-  (I  +  A)jr3  +  1.9  =  o, 

(f  +  I)  A'.  +  (I  +  5  +  I  +  i)  a;  +  8.0  =  o, 

(I  +  t\)  A'.  +  (I  +  i  +  f  +  ii)K,  +  8.6  =  o. 

and  their  solution  gives  the  values 

A',  =  +6.99,     a;  =  -7-53.     A'3=  -9.43» 
from  which  the  followinsf  corrections  are  found : 


w^  =  -0  .5, 

Xr  = 

-0.4, 

yi  = 

-2.5, 

z,  =  -4.1, 

W,=    -    O.I, 

X,  = 

-5-0, 

3-2  = 

-3-1, 

Zz  =    —0.9. 

7'he  final  adjusted  values  of  the  single  angles  now  are 

IV,  =    64    08    35.9,  K=    53    53   45.9, 

^1  =    36    34    19-6,  ^i  =    46    49    09-6, 

^2=    29    59    39.0,  Z2  =    37    18    16.0. 

The  adjusted  values  of  the  large  angles  are  now  obtained  by 
simple  addition  of  the  single  angles,  and  are 


PV  = 

106   07  24  .4, 

r=  103°  ii'ii".4, 

X  = 

66    2,1    58.6, 

Z  =      84   07    25.6, 

whose  sum  is  exactly  360  degrees. 

121.  In  geodetic  surveys  where  the  sides  of  the  quadrilateral 
are  many  miles  in  length,  the  spherical  excess  must  be  con- 
sidered in  stating  the  conditional  equations  for  the  three  tri- 
angles. In  such  work  a  fourth  conditional  equation  must  also 
be  introduced  in  order  to  insure  that  the  length  of  any  side 
shall  be  the  same  through  whatever  set  of  triangles  it  be  com- 
puted. The  development  of  the  calculations  for  such  cases 
belongs  properly  to  works  on  geodesy,  and  will  not  here  be  dis- 
cussed.     Detailed   examples   of   the  method   may   be  seen  in 


152  CONDITIONED    OBSERVATIONS.  VIII. 

Schott's  article  on  the  adjustment  of  the  horizontal  angles  of  a 
triangulation  in  the  United  States  Coast  Survey  Report  for 
1854,  in  Clarke's  Geodesy  (Oxford,  1880),  and  in  many  German 
works  on  higher  surveying.* 


Simple  Triangjilation. 

\ii.  In  the  adjustment  of  a  simple  triangulation  the  method 
of  procedure  is  essentially  the  same  as  for  a  quadrilateral. 
First,  the  adjustment  of  the  angles  at  each  station  should  be 
made,  and  then  the  resulting  values  further  corrected,  so  as 
to  satisfy  the  geometrical  requirements  of  the  figure.  This 
method  is  not  strictly  in  accordance  with  the  fundamental 
principle  of  Least  Squares.  By  the  station  adjustment  a  cor- 
rection, 7\,  is  found  for  each  angle,  and  by  the  figure  adjustment 
another  correction,  i\ ;  so  that  the  total  correction  is  v^  -\-  Vj. 
The  fundamental  principle  for  observations  of  equal  weight 
requires  that  ^{v,  -f-  z'^)-  should  be  made  a  minimum  in  order  to 
obtain  the  best  values  of  the  corrections,  while  by  the  method 

pursued  2■^^^  is  made  a  minimum 
in  the  first  adjustment,  and  ^t/ 
a  minimum  in  the  second.  The 
reason  for  deviating  from  the 
strict  letter  of  the  law  is,  that 
the  general  method  of  determin- 
ing the  total  equation  at  once  is 
too  laborious,  owing  to  the  large 
number  of  conditional  equations  involved.  Usually  also  the 
difference  between  the  final  results  of  the  two  methods  will  be 
small.  In  the  next  article  will  be  given  a  comparison  of  the 
two  methods  as  applied  to  a  simple  case. 


*  See    also    Merriman's    Elements    of    Precise    Surveying  and   Geodesy. 
New  York,  1899. 


§  123. 


SIMPLE    TRIANGULATION. 


153 


123.  The  following  observations  were  made  to  determine 
the  distance  between  the  non-intervisible  stations  C  and  D 
by  means  of  a  measured  base  AB : 


BAC  = 

2f 

09 

o5"-5, 

BAD^ 

51 

34 

35-5, 

CAD  = 

24 

25 

27.8, 

ABD^ 

70 

08 

32.1, 

ABC  = 

128 

29 

07-5, 

DBC  = 

58 

20 

38.4, 

ACB  = 

24 

21 

46.0, 

ADB^ 

58 

16 

50.8. 

By  the  strict  method  of  Art.  113  or  Art.  114  the  four  condi 
tional  observations  are  written,  one  for  each  of  the  points  A 
and  B,  and  one  for  each  triangle,  and  the  adjusted  values  found 
as  given  in  the  second  column  of  the  following  table  : 


Observed. 

Adjusted. 

V. 

^/^ 

o5"-5 

06.2 

+  0.7 

0.49 

35-5 

35-6 

+   O.I 

O.OI 

27.8 

29.4 

+  1.6 

2.56 

32.1 

32.0 

—  0.1 

O.OI 

07-5 

08.6 

+  I.I 

1. 21 

38.4 

36.6 

-1.8 

3-24 

46.0 

45-2 

-0.8 

0.64 

50.8 

524 

+  1.6 

2.56 

The  sum  %v^  is  here  10.72,  and  by  (34)  the  probable  error  of  a 
single  observation  is 


^       ,  /10.72        „ 
r=  0.6745V/ =  I  .1. 

T  A 


154 


CONDITIONED    OBSERVA  -r/ONS. 


VIIl. 


By  the  shorter  method  the  local  adjustment  at  A  and  B  is 
first  made,  giving  the  results 

BAC  =     27°  09' o6".2,  weight  1.5, 

BAD=    51  34     34.S,  weight  1.5. 

ABD  =     70  08     31. 1,  weight  1.5, 

ABC  =  128  29     08.5,  weight  1.5. 

The  triangles  ABC  and   BAD  are  next   separately  adjusted, 
using  these  four  angles  and  those  at  C  and  D.     The  results  are 


Observed. 

Adjusted. 

Z'. 

v\ 

o5"-5 

06.0 

+  0.5 

0.25 

35 

5 

35-7 

+  0.2 

0.04 

27 

8 

29.7 

+  1.9 

3.61 

32 

I 

32.0 

—  0.1 

O.OI 

07 

5 

08.3 

+  0.8 

0.64 

38 

4 

^('■Z 

—  2.1 

4.41 

46 

0 

45-7 

-  0.3 

0.09 

50 

8 

52-3 

+  1-5 

2.25 

The  sum  St'Ms  here  11.3,  which  is  but  slightly  greater  than 
that  of  the  stricter  method.  A  comparison  of  the  two  sets  of 
adjusted  values  shows  also  that  the  differences  are  small. 


Levelling. 

124.  A  simple  discussion  of  the  precision  of  levelling  observa- 
tions involving  but  one  conditional  equation  will  here  be  given 
as  an  illustration  of  the  general  method  of  treatment  of  Art.  113. 

There  are  three  points,  A,  B,  and  C,  situated  at  nearly  equa'i 
distances  apart,  but  upon  different  levels.     In  order  to  ascertain 


§  124. 


LEVELLING. 


155 


with  accuracy  th^ir  relative  heights,  a  levelHng  instrument  was 
set  up  between  A  and  B,  and  readings  taken  upon  a  rod  held 
at  those  points,  with  the  results. 

On  rod  at  A,   8.7342  feet,  mean  of  12  readings. 
On  rod  at  B,  2.3671   feet,  mean  of  9  readings. 

The  instrument  was  then  moved  to  a  point  between  B  and  C, 
and  the  observations  taken. 

On  rod  at  B,     5.0247  feet,  mean  of  7  readings, 
On  rod  at   C,   11.2069  feet,  mean  of  4  readings. 

Lastly,  the  level  was  set  up  between  C  and  A,  and  the  rods 
observed. 

On  rod  at   C,  0.4672  feet,  mean  of  5  readings, 
On  rod  at  A,  0.6510  feet,  mean  of  3  readings. 

It  is  required  to  find  the  adjusted  values  of  these  readings,  the 
most  probable  differences  of  level  between  the  points,  and  the 
probable  error  of  a  single  reading  on  the  rod. 

First  arrange  these  measurements  as  they  would  be  written 
in  an  engineer's  level-book,  and,  assuming  the  elevation  of  A 
as  0.0,  find  the  heights  of  the  other  points. 


Station. 

Back  Sight. 

Fore  Sight. 

Height  of 
Instrument. 

Elevation 
above  A. 

,B, 

^3 

8.7342 

5-0247 
0.4672 

2.3671 

11.2069 

0.6510 

8.7342 

II.3918 

0.6521 

0.0 

6.3671 
0.1849 
O.OOI  I 

The  number  of  readings  or  the  weight  of  each  sight  is  placed 
in  the  first  column  preceding  and  following  the  name  of  the 


156  CONDITIONED   OBSERVATIONS.  VIII. 

station  ;  thus  ^B^  denotes  that  the  back  sight  on  B  has  a  weight 
of  7,  and  the  fore  sight  one  of  9.  Regarding  the  elevation  of 
A  as  o,  that  of  B  comes  out  6.3671  feet,  that  of  C,  0.1849  ^^^t  ; 
and,  on  returning  to  the  starting-point,  it  is  found  that  A  is 
o.ooii  feet,  instead  of  o  as  it  ought  to  be. 

Represent  the  back  sights  upon  A,  B,  and  C  by  Z^,  Z^,  and 
Z^,  and  the  fore  sights  upon  B,  C,  and  A  by  Z^,  Z^,  and  Z^, 
and  let  ;:,,  c^,  s^,  .c,,  c^,  and  S(,  be  corrections  to  be  applied  to 
those  observed  values.     The  observation  equations  then  are 


2j  =  0,  weight 

12, 

z^  =  0,  weight  9, 

z^  =  0,  weight 

7> 

z^  =  0,  weight  4, 

25  =  0,  weight 

5> 

26  =  0,  weight  3, 

and  the  conditional  equation  is 

Zi  +  z^  -\-  z^  —  Z2  —  z^  —  Ze  =  —  O.OOII. 

From  the  conditional  equation  take  the  value  of  ^^,  and  insert 
it  in  the  observation  equations,  which,  after  multiplication  by 
the  square  roots  of  their  respective  weights,  become 

yT^z,  =  o, 

)/s  25  =  o, 
3  S2  =  o, 

V/3  C6  =  O, 
22i   +  2^3  +   205  —    22,  —    2Z(,  =   —  0.0022. 

From  these  the  normal  equations  (Art.  48)  are 

162,  +    423  +  425  —    42^  —  4^6  =  —  0.0044, 

42,  +  II23  +  4Z5  —    422  —  425  =  —  0.0044, 

42,  -f    423  4-  925  —    42,  —  425  —  —  0.0044, 

—  42,  -     423  -425+1  ;^s,  +  420  =  +  0.0044, 

—  4s.  —     423  —  425  4-     422  +  725  =  +  0.0044, 


§  125- 


LEVELLING. 


157 


the  first  being  the  normal  equation  for  z„  the  second  for  ^,,  the 
third  for  ^5,  the  fourth  for  z^,  and  the  fifth  for  Z(,.  The  solu- 
tion gives  the  following  results  : 

Zi=  —  0.00008,        S3  =  —  0.00014,        ^5  =  ~  0.00020,  1 

22  =    +   O.OOOII,  S4  =    +  0.00024,  26  =    +  0.00033. 

Applying  these  to  the  observed  values,  the  adjusted  results  are 


Station. 

Back  Sight. 

Fore  Sight. 

Elevation 
above  A. 

A 
B 
C 
A 

8.73412 
5.02456 
0.46700 

2.36721 
II. 20714 

0-65133 

0.0 

6.36691 
0.18433 
0.0 

The  residuals  are  in  this  case  the  corrections  z„  z^,  etc. 
Squaring  these,  multiplying  each  square  by  its  weight,  and  add- 
ing, gives 

"^pv-  =  0.000001079. 
From  formula  (34)  then 

r  —  o.6745y/o. 000001079  =  0.0007  f^et,   ' 
which  is  the  probable  error  of  a  single  reading  on  the  rod. 

125.  The  adjustment  of  a  network  of  level  lines  may  also 
be  effected  by  the  method  of  conditioned  observations.  When 
the  levelling  is  of  the  same  precision  throughout,  the  probable 
errors  of  differences  of  level  should  be  taken  as  varying  with 
the  square  root  of  the  lengths  of  lines,  being  governed,  in  short, 


158 


CONDITIONED    OBSERVATIONS. 


VIII. 


by  the  same  law  of  propagation  of  error  as  linear  measure- 
ments (see  Art.  91).  Each  difference 
of  level  should  hence  be  assigned  a 
weight  inversely  proportional  to  the 
length  of  the  line  between  the  two 
points.  For  each  triangle  or  polygon 
of  the  network,  there  is  the  rigorous 
condition  that  the  sum  of  the  difter- 
ences  of  level  shall  be  zero.  From 
these  conditional  equations,  corrections 
to  the  observed  differences  of  level  are 
determined  by  the  method  of  Art.  114. 

As  an  example,  consider  the  follow- 
ing eight  differences  of  level  forming 
three  closed  figures,  ABE,  BCFE,  and 
CDF: 


No. 

Stations. 

Diff.  Level. 

Distance. 

Weight. 

Feet. 

Miles. 

I 

B  above  A 

120.2 

4.0 

0.25 

2 

C  above  B 

230.6 

7.2 

0.14 

3 

D  above  C 

143.0 

5-0 

0.20 

4 

D  above  F 

294.4 

6.3 

0.16 

5 

C  above  F 

150.2 

2.0 

0.50 

6 

^ above  E 

934 

4.8 

0.21 

7 

B  above  E 

14-5 

3-5 

0.29 

8 

E  above  A 

106.7 

^■l 

0.12 

It  is  required  to  find  the  most  probable  corrections  to  the  above 
differences  of  level  in  order  to  cause  the  discrepancies  in  the 
three  polygons  to  vanish 


§125.  LEVELLING.  159 

Let  /-„  7/2,  etc.,  represent  the  most  probable  differences  of 
level.     Then  the  three  conditions  are 


for  ABE, 

//,  —  h^  —  /?s  =  o, 

{ox  BCFE, 

K 

—  //5    —   //6   +  h-j  =   O, 

for  CDF, 

h^  -  h^  +  h^  =  0. 

Let  z'„  v^,  etc.,  be  the  most  probable  corrections  to  the  observed 
differences  of  level,  so  that 

hy  =  I20.2  +  i\,        /h  =  230.6  +  V2,        etc. 
Then  the  three  conditional  equations  become 

V^  —  ?';  —  Z'8  —    I-O  =  O, 

V2  —  z's  —  7>6  4-  e'7  +  ^-5  =  o» 

Z'3  —  Z'4  4-  Z'j  —   1.2  =  o. 

From  these  the  correlative  equations  are  written,  the  weight 
of  each  v  being  taken  as  the  reciprocal  of  the  corresponding 
distance : 

t\  =  +  4.0A',, 
?'2  =  +  7-2A'2, 

^'3   =    +    5   0^3» 

^'4  =  —  6.3A'3, 

z'j  =  —  2.0K2  +  s.oA'j, 

z'6  =  —  4-8A'„ 

7'7  =  -  3-5^'.  +  3-5^2, 
Vi  =  —  8. 3  A',. 

Next  the  three  normal  equations  are 

15. 8A',  —    3-5^'2  —  i-o  =  Oj 

—  3..sA^  +  i7-5^'2  —     2.oA'3  +  1.5  =  o, 

—  2.0A',  -f  i3.3A'3  —  1.2—0, 

and  the  solution  of  these  gives 

A',  =  +  0.04848,         A'2  =  —  0.066855,         A'3  =  +  0.08017 


i6o 


CONDITIONED    OBSER  \  'A  TIONS. 


VIII. 


Lastly,  by  substituting  these  in  the  correlative  equations,  the 
corrections  are  found,  which  are  given  in  the  third  column  of 
the  following  table,  while  in  the  fourth  are  the  adjusted  results. 


No. 

Observed 
Diff.  Level. 

V. 

Adjusted 
Diff.  Level. 

I 

I20.2 

+  0.19 

120.39 

2 

230.6 

—  0.48 

230. 1  2 

3 

143.0 

+  0.40 

143.40 

4 

294.4 

-0.51 

293.89 

5 

150.2 

+  0.29 

150-49 

6 

934 

+  0-32 

93  72 

7 

14-5 

—  0.40 

14.10 

8 

106.7 

—  0.40 

106.30 

In  order  to  ascertain  the  precision  of  the  work,  the  correc- 
tions are  squared,  and  each  square  multiplied  by  its  respective 
weight,  and  the  sums  of  these  products  taken.  This  sum  is 
about  0.246;  and  then  by  (34)  the  probable  error  of  an  obser- 
vation of  the  weight  unity,  that  is,  the  probable  error  of  the 
difference  of  level  of  the  ends  of  a  line  one  mile  in  length,  is 


a  result  that  indicates  a  low  degree  of  precision. 


126.   Problems. 

I.  Adjust  the  following  angles  taken  at  the  station  O : 

A  OB  -    40°  52'   37".     weight  16, 
BOC  z=    92     25     41.      weight  4, 
COD-    So      6     15,      weight  3, 
DOA  -  146     35     20,      weight  i. 


§  126.  PROBLEMS.  l6l 

2.  In  a  spherical  triangle  XYZ  the  three  measured  angles  are 

A"  =93°  48'  i5".22,  with  weight  30, 
^=  5'  55  o.\f>,  with  weight  19, 
Z  =  34     16     49.72,     with  weight  13. 

The  spherical  excess  is  4".05.     What  are  the  adjusted  angles? 

3.  In  a  quadrilateral  WXYZ,  the  following  angles,  all  of  equal  weight, 

are  measured,  and  it  is  required  to  adjust  them. 

IV  -  106°  07'  30",  Fi  =  49°  17'  23", 

W^-    41     58    47,  y^-^i    53  50, 

W2.  —    64    08     34,  Z   =  84    07  18, 

A'  =    66    34    09,  Z2  =  '^-]     18  12. 

-^i  =    36    34     21, 

4.  Aajust  the  level  observations  in  Art.  100  by  the  method  of  condi- 
tioned observations,  taking  the  weights  as  equal. 

5.  Discuss  the  method  of  correcting  the  latitudes  and  departures  in 
a  compass  survey  of  a  field. 

6.  Two  bases,  AB  and  DE,  are  connected  by  three  triangles,  ABC, 
BCD,  and  CDE.  The  bases  are  measured,  and  also  the  three  angles 
of  each  triangle.  State  the  four  conditional  equations,  and  explain  in 
detail  the  process  of  adjustment. 


102  THE  DISCUSSION  OF  OBSERVATIONS.  IX. 


CHAPTER    IX. 

THE   DISCUSSION   OF    OBSERVATIONS. 

127.  In  the  preceding  pages  it  has  been  shown  how  to  adjust 
observations,  and  how  to  ascertain  their  precision  by  means  of 
the  probable  error.  By  thus  treating  series  or  sets  of  measure- 
ments, a  comparison  or  discussion  may  be  instituted  concern- 
ing the  relative  degrees  of  precision,  the  presence  of  constant 
errors,  and  the  best  way  to  improve  the  methods  of  observa- 
tion. In  this  chapter  it  is  proposed  to  present  some  further 
remarks  relating  to  the  discussion  ot  observations  by  the  use  of 
the  fundamental  law  of  probability  of  error,  and  to  indicate  that 
this  law  is  also  applicable  to  social  statistics,  and  that  it  really 
governs  the  way  in  which  the  laws  of  nature  are  executed. 


Probability  of  Errors. 

128.  In  Chap.  II  a  method  of  investigating  the  probability  of 
errors,  and  comparing  theory  with  experience,  was  given,  in 
which  it  was  necessary  to  assume  the  value  of  the  measure  of 
precision  h.  For  instance,  in  Arts.  19  and  33  there  are  dis- 
cussed one  hundred  residual  errors,  for  which  the  value  of  h  is 
stated  to  be  ~ — .  It  is  now  easy  to  see  that  this  value  may 
be  found  at  once  from  the  probable  error  r  by  means  o*^  the 
formula  (17),   while  r  is  deduced  from  the  formula  (20).     Tp 


§  128.  PROBABILITY  OF  ERRORS.  163 

compare,  then,  the  theoretical  and  actual  distribution  of  errors 
for  such  cases  by  the  use  of  Table  I  it  is  only  necessary  to 
deduce  the  value  of  r  in  the  usual  way,  and  from  it  to  find  h, 
which  enters  as  an  argument  in  the  table. 

It  is  evident,  then,  that,  in  undertaking  such  discussions,  it  h 
more  convenient  to  have  a  table  of  the  values  of  the  probability 
integral  in  terms  of  r  as  an  argument.     Such  is  Table  II  at 

the  end  of  this  book,  which  gives,  for  successive  values  of  -,  the 

r 
probability  that  a  given  error  is  less  numerically  than  x,  or  that 

it  lies  between  the  limits  — ;tr  and  -\- x. 

To  illustrate  the  use  of  Table  II  consider  an  angle  for  which 
the  mean  value  is  found  to  be 

37°    42'     i3".92  ±  o".25. 

Now,  from  the  definition  of  probable  error,  it  is  known  that  the 
probability  is  ;  that  the  actual  error  of  the  result  is  less  than 
o".25.  Let  it  be  asked  what  are  the  respective  probabilities  that 
the  actual  error  is  less  than  the  amounts  o".5  and  i" .o.  Trom 
the  table 

for  -  =  -^  =  2,  P  =  0.823, 

r       0.25 

X        1. 00 

Hence  the  probability  that  the  error  in  the  result  is  less  than 
o".5  is  0.823,  or  it  is  a  fair  wager  of  823  to  177  that  such  is 
the  case.  And  the  probability  that  the  error  is  less  than  i".o 
is  0.993,  or  it  is  a  fair  wager  of  993  to  7  that  such  is  the 
case. 

As  the  number  of  errors  is  proportional  to  the  probability, 
the  values  of  the  integral  need  only  to  be  multiplied  by  the 
total  number  of  errors  to  give  the  theoretical  number  less  than 


164  THE  DISCUSSION  OF  OBSERVATIONS.  IX. 

certain  limits.  For  example,  in  one  thousand  errors  or  residu- 
als, there  should  be 

264  less  than  \r,  and  736  greater, 

500  less  than    r,  and  500  greater, 

823  less  than  2/-,  and  177  greater, 

957  less  than  3;',  and    43  greater, 

993  less  than  4;-,  and       7  greater, 

999  less  than  5;-,  and       i  greater. 

Table  II  gives  only  four  decimal  places,  which  suffice  for 
any  ordinary,  investigation.  By  the  methods  of  calculation 
explained  in  Chap.  II  more  decimals  may  be  deduced,  and  the 
following,  results  be  found  for  the  theoretical  distribution  of 
errors  when  the  total  number  of  errors  is  one  hundred  thou- 
sand : 

95698  are  less  than  3/-,  and  4302  greater, 

99302  are  less  than  4;-,  and  698  greater, 
99926  are  less  than  5;',  and  74  greater, 
99995  are  less  than  6r,  and        5  greater. 

As  the  frequency  with  which  an  error  occurs  is  expressed  by 
its  probability,  it  is  evident  that  errors  greater  than  five  or  six 
times  the  probable  error  should  be  very  rare. 

129.   As  shown  in  Art.  35,  the  probability  of  the  error  o  is 
—^r,  or,  introducing  for  h  its  value  ,  it  may  be  written 

V/tt  ^ 

dx 
jo=  0.2691  — . 
r 

Here  dx  is  the  interval  between  successive  values  of  x.  It 
there  be  N  errors  in  a  series,  the  number  having  the  value  0 
should  hence  be 

(42)  iVo  =  0.2691  — N, 

r 

where  r  is  the  probable  error  of  a  single  observation. 


§  129. 


rKOBABILITY  OF  ERRORS. 


165 


Formula  (42)  affords  a  rough  comparison  of  theory  and  ex- 
perience without  the  use  of  tables.  For  instance,  let  the 
target-shots  described  in  Art.  18  be  again  considered,  and 
regard  those  in  the  middle  division  as  having  the  error  o,  those 
in  the  next  division  above  as  having  the  error  -f-  i,  and  so 
on.  Then  the  errors,  without  regard  to  sign,  are  as  in  the  first 
column  below,  their  squares  in  the  second,  their  weights  or  the 
number  of  shots  in  the  third,  and  the  weighted  squares  in  the 
fourth. 


X. 

X-. 

/• 

/x=. 

/■ 

0 

0 

212 

0 

261 

I 

I 

394 

394 

382 

2 

4 

282 

1,128 

232 

3 

9 

S9 

801 

93 

4 

16 

20 

320 

26 

5 

25 

3 

75 

6 

2/.V2  = 

=  2,718 

1,000 

Now,  the  probable  error  of  a  single  observation  is 


\  1000 


and,  by  formula  (42),  the  number  of  errors  having  the  value  O  is 
,,       0.2691  X  I  X  1000 

^o  =   • =    245, 


I.I 


which  is  a  satisfactory  agreement  with  the  actual  number  212. 
In  the  last  column  of  the  above  table  are  given  the  theoretical 
numbers  of  errors  as  computed  from  Table  II. 


1 66  THE  DISCUSSION  OF  OBSERVATIONS.  IX, 

TJie  Rejection  of  Doubtful  Observatiofts. 

130.  The  theoretical  distribution  of  errors,  according  to  the 
fundamental  formula  (i),  is  shown  by  the  values  of  the  proba- 
bility integral  given  in  Table  II  ;  and  from  these  it  is  seen,  as 
in  Art.  128,  that  the  number  of  errors  greater  than  4r  or  5;' is 
very  small.  It  becomes,  then,  a  question,  whether  the  probabil- 
ity of  an  error  might  not  be  so  small  that  it  would  be  justifiable 
to  reject  entirely  the  corresponding  observation.  For  instance, 
if  one  thousand  direct  observations  be  taken,  the  probability 
that  there  will  be  one  error  greater  than  5;' is  ^;  if,  then,  in 
taking  a  series  of,  say,  fifty  observations,  one  error  should  exceed 
5;',  the  probability  of  its  occurrence  would  be  very  much  smaller 

than  ~^,  and  the  observer  would  be  tempted   to  reject  that 
1000'  r  J 

observation.  But  undoubtedly  it  would  be  a  dangerous  thing 
to  allow  an  observer  to  decide  upon  his  own  limit  of  rejection. 
It  has  accordingly  been  proposed  to  attempt  to  establish  a  cri- 
terion by  which  the  limit  may  be  legitimately  established  from 
the  principles  of  the  probability  of  error.  The  critepon  pro- 
posed by  Chauvenet  is  the  simplest  of  those  deduced  for  this 
purpose,  and  is  the  following  : 

Let  n  be  the  number  of  direct  observations,  and  also  the 
number  of  errors.  Let  r  denote  the  probable  error  oi  a  single 
observation    as   found  from  the  n  residuals   by  forp^-ula   (20). 

Let  X  be  the  limiting  error,  and  let  -  be  called  t.     Le^  P  be  the 

r 

value  of  the  integral  in  Table  II  corresponding  to  /.      Then 
(43)  P=  — — ,      and     X  =  tr 

is  the  criterion  for  the  rejection  of  the  largest  residual 

To  prove  this,  consider  that  the  quantities  in  Table  II  need 
only  be  multiplied  by  the  total  number  of  errors  to  show  the 
actual  distribution ;  so  that  nP  indicates  the  number  of  «r»-or£ 


§131- 


REJECTIOX  OF  DOUBTFUL    OBSERVATIONS. 


i6y 


less  than  x,  and  ;/  —  nP  indicates  the  number  greater  than  x. 
Now,  if 


n  —  nP=l 


there  is  but  half  an  error  greater  than  .r,  and  any  error  greater 
than  this  x  would  be  larger  than  allowed  by  the  theoretical  dis- 
tribution. Hence  the  value  of  x  corresponding  to  this  value  of 
P  is  the  limiting  value,  which  indicates  whether  the  greatest 
residual  in  a  series  may  be  rejected  or  not. 

131.  In  order  to  facilitate  the  use  of  this  criterion,  Table  VII 
has  been  computed,  giving  the  value  of  /  directly  for  several 

values  of  n.      For  instance,  if  n  is  5,  the  value  of  P  is  , 

10 

or  0.9;  and  from  Table  VI 1  the  corresponding  value  of  /  is  2.44. 

The  following  particular  example  will  illustrate  the  method 
of  procedure.  Let  there  be  given  thirteen  observations  of  an 
angle,  as  in  the  first  column  below. 


62°    i2'5i".75 

2.69 

7.24 

48.45 

0.61 

0-37 

50.60 

1-54 

2-37 

47-85 

1. 21 

1.46 

5^-05 

1.99 

3-96 

47-75 

1-31 

1.72 

47.40 

1.66 

2.76 

48.85 

0.21 

0.04 

49.20 

0.14 

0.02 

48.90 

0.16 

0.03 

50-95 

1.89 

3-57 

50-55 

1-49 

2.22 

44-45 

4.61 

21.25 

62°    i2'49".o6 

47.01 

I 

1 68  THE   DISCUSSION   OF   OBSERVATIONS.  IX. 

Let  the  mean  of  these  be  found,  the  residuals  placed  in  the 
second  column,  and  their  squares  in  the  third.  The  sum  Iv^ 
is  47.01  ;  and  hence,  from  (20),  tlie  probable  error  r  of  a  single 
observation  is  i".32.  Table  VII  gives  /  =  3.07  when  ;/  =  13  : 
hence,  by  the  criterion,  the  limiting  error  is 

X  =  3.07  X  1.32  =  4-05. 

and  accordingly  the  largest  residual  4.61  should  be  rejected. 
To  ascertain  if  the  next  largest  residual,  2.99,  should  aiSD  be 
rejected,  the  mean  of  the  twelve  good  observations  should  be 
found,  and  a  new  r  computed  from  the  twelve  new  residuals. 
But  evidently  the  new  sum  ^v^  will  not  differ  greatly  from  the 
former  sum  minus  the  square  of  the  rejected  residual,  or 
new  Iv"-  =  47-OI  —  21.25  —  25.76, 

from  which  the  new  r  is  found  to  be  about  i".03.  Then  the 
limiting  error  is 

A-=  3.02  X  1.03  =:  3".ii, 

which  shows  that  the  residual  2.99  is  not  to  be  rejected. 

132.  Hagen's  deduction  of  the  law  of  probability  of  error, 
given  in  Chap.  II,  suggests  another  method  of  finding  the 
limiting  error  of  observation,  and  a  new  criterion  for  rejection. 
In  Art.  26  the  maximum  error  is  expressed  by  ;«A,r,  and  the 

quantity  m^x^  is  replaced  by  —7^.     It  is  hence  easy,  by  the  help 

of  (17),  to  find 

(44)  ff^^x  =  4.4  ^» 

where  dx  is  the  constant  interval  between  successive  values  of 
the  errors.  For  the  observations  discussed  in  Art.  129  this 
formula  gives  the  limiting  error  in\x  as  5.3,  which  seems 
entirely  satisfactory.  It  is  not  possible  to  apply  it,  however,  to 
angle  measurements  like  those  of  the  last  article,  on  account  of 
the  impossibility  of  assigning  a  proper  value  to  the  interval  dx. 


s 


133.  CONSTAN'T  ERRORS.  1 69 


The  same  difficulty  prevents  the  practical  use  of  formula  (42), 
except  in  cases  where  this  constant  interval  is  definitely  known. 

There  is  another  criterion,  due  to  Peirce,  which  may  be 
applied  to  the  case  of  indirect  observations  involving  several 
unknown  quantities,  as  well  as  to  that  of  direct  measurements  ; 
but  its  development  cannot  be  given  here.  In  general,  it  should 
be  borne  in  mind  that  the  rejection  of  measurements  for  the 
single  reason  of  discordance  with  others  is  not  usually  justi- 
fiable unless  that  discordance  is  considerably  more  than  indi- 
cated by  the  criterions.  A  mistake  is  to  be  rejected,  and  an 
observation  giving  a  residual  greater  than  ^r  or  5;'  is  to  be 
regarded  with  suspicion,  and  be  certainly  rejected  if  the  note- 
book shows  any  thing  unfavorable  in  the  circumstances  under 
which  it  was  taken.  Usually,  in  practice,  the  number  of  large 
errors  is  greater  than  should  be  the  case,  according  to  theory ; 
and  this  seems  to  indicate,  either  that  the  series  is  not  suf- 
ficiently extended  to  give  a  reliable  value  of  r,  or  that  abnormal 
causes  of  error  affect  certain  observations.  If  it  were  possible 
to  increase  the  number  of  measurements,  it  would  undoubtedly 
be  found  that  the  abnormal  errors  would  be  as  often  positive  as 
negative,  and  that,  for  a  very  great  number,  there  would  be  few 
that  could  be  rejected  by  the  criterion. 

Constant  Errors. 

133.  In  all  that  has  preceded,  it  has  been  supposed  that 
the  constant  errors  of  observation  have  been  eliminated  from  the 
numerical  results  before  discussing  them  by  the  Method  of 
Least  Squares.  If  this  is  not  done,  and  all  the  measurements 
of  a  set  are  affected  by  the  same  constant  error,  that  error 
will  also  appear  in  the  adjusted  result.  For  instance,  suppose 
thirty  shots  to  be  fired  with  the  intention  of  hitting  the  centre 
of  a  target,  and  let  their  actual  distribution  be  as  shown  in  the 
figure.  The  most  probable  location  of  the  centre,  according  to 
the  records,  is  about  two  spaces  to  the  right,  and  about  half  a 


I/O 


THE  DISCUSSION  OF  OBSERVATIONS. 


IX. 


Fig. 

13. 

• 

• 

\ 

• 

%' 

• 
• 

• 

%, 

■:\ 

• 

• 
• 

• 

»• 

• 

• 

• 

space  below  the  true  centre.  Each  shot,  then,  has  been  subject 
to  these  constant  errors  ;  the  first  due,  perhaps,  to  the  wind, 
and  the  second  to  gravity.     If,  now,  these  marks  on  the  target 

represented  observations  for 
the  purpose  of  locating  the 
centre,  the  result  obtained  by 
their  adjustment  would  be  in. 
error  by  the  amounts  just 
stated.  Therefore,  if  all  the 
observations  of  a  series  are 
affected  by  the  same  constant 
error,  the  Method  of  Least 
Squares  can  do  nothing  but 
adjust  the  accidental  errors; 
and  the  probable  errors  of  the 
adjusted  results  refer  only  to 
them,  and  give  no  indication 
whether  constant  causes  of  error  affect  the  measurements  or  not. 

134.  The  probability  of  the  existence  of  a  constant  error  in 
a  case  like  that  just  illustrated  is  evidently  large,  and  the 
numerical  probability  of  its  value  lying  between  certain  limits 
may  be  found  by  the  help  of  Table  II.  The  following  is  an 
example  of  such  a  discussion  : 

Suppose  that  an  angle  is  laid  out  with  very  accurate  instru- 
ments, and  tested  in  many  ways,  so  that  its  true  value  may  be 
regarded  as  exactly  90°.  Let  twenty-five  observations  be  taken 
upon  it  with  a  transit  whose  accuracy  is  to  be  tested,  and  let 
the  mean  of  those  measurements  be  89°  59'  57"  ±  o".8.  Then 
it  is  extremely  probable  that  a  constant  error  of  about  —  3" 
exists  in  the  instrument.  To  find  the  numerical  expression  of 
this  probability,  suppose  that  the  true  value  of  the  angle  was 
unknown,  and  ask  the  probability  that  the  mean  is  within  2"  of 

the  truth.     Then,  for  -■  r=  —  =  2.5,  the  value  of  the  integral  in 

r       0.8 


§  135.  CONSTANT  ER-RORS.  I /I 

Table  II  is  0.908  ;  so  that  it  is  a  wager  of  908  to  92,  or  of 
almost  10  to  I,  that  the  mean  is  between  the  limits  89°  59'  55" 
and  89°  59'  59".  Hence,  since  the  angle  is  known  to  be  90°,  it 
must  be  the  same  probability  and  the  same  wager  that  there 
is  a  constant  error  lying  between  the  limits  —  \"  and  —  5". 
So,  also,  if  X  =  3",  it  may  be  shown  that  it  is  a  wager  of  39  to  i 
that  there  is  a  constant  error  between  o"  and  —  6". 

135.  In  case  that  several  sources  of  constant  error  exist,  the 
adjustment  by  the  Method  of  Least  Squares  tends  to  elimi- 
nate them,  and  to  give  results  nearer  and  nearer  to  the  actual 
values,  as  the  number  of  observations  is  increased.  This  will 
be  rendered  evident  by  considering  again  the  illustration  of 
che  target.  One  marksman  fires  thirty  balls,  which  are  subject 
to  a  constant  error,  as  in  Fig.  13.  Another  marksman  fires 
thirty  more,  which  have  a  different  constant  error,  owing  to  the 
peculiarities  in  his  aim.  A  third  marksman  has  a  third  con- 
stant error,  in  a  still  different  direction.  The  shots  of  each 
marksman  are  distributed  around  their  most  probable  centre 
in  accordance  with  the  law  of  probability  of  accidental  errors. 
And  undoubtedly  these  constant  errors  will  be  grouped  around 
the  true  centre  according  to  the  same  law  ;  and,  as  the  number 
of  marksmen  increases,  the  constant  errors  will  thus  tend  to 
annul  each  other,  and  ultimately  make  the  most  probable  centre 
coincide  with  the  true  one. 

And  so  it  must  be  in  angle  observations,  when  great  pre- 
cision is  demanded.  On  one  day  certain  constant  errors,  due 
to  atmospheric  influences,  affect  all  results  in  a  certain  direc- 
tion ;  while  on  a  second  day,  under  different  influences,  new 
constant  errors  act  in  another  direction.  If  the  measurements 
be  continued  over  many  days,  the  number  and  magnitude  of 
positive  constant  errors  will  be  likely  to  equal  the  negative 
ones  ;  so  that  the  adjustment  by  the  Method  of  Least  Squares 
will  balance  them,  and  give  results   near  to  the  true  values. 


172 


THE   DISCUSSION  OF  OBSERVATIONS. 


IX. 


Here  may  be  seen  the  reason  why  the  number  of  large  residuals 
is  usually  greater  than  the  theory  demands,  ana  also  a  reason 
why  a  criterion  for  rejection  cannot  generally  be  safely  applied 
to  series  of  observations  consisting  of  few  measurements. 


Social  Statistics. 

136.  It  is  found  that  the  law  of  probability  of  error  applies 
to  many  phenomena  of  social  and  political  science.  If  men 
be  arranged  in  groups,  according  to  their  heights,  there  will  be 
found  few  dwarfs  and  few  giants ;  and  the  numbers  in  the  dif- 
ferent groups  will  closely  agree  with  the  theoretical  distribu- 
tion required  by  the  curve  of  probability.  The  following  table, 
which    is    taken    from    Gould's    Statistics    (New    York,    1869), 


Height. 
Inches. 

Actual 
Number. 

Proportional  Number  in  10,000. 

Observed. 

Calculated. 

Calc  — Obs. 

61 

197 

105 

100 

-     5 

62 

317 

169 

171 

+      2 

63 

692 

369 

368 

—     I 

64 

1289 

686 

675 

—  1 1 

65 

1961 

1044 

1051 

+    7 

66 

2613 

1391 

1399 

-f    8 

67 

2974 

1584 

1584 

0 

68 

3017 

1607 

1531 

-=76 

69 

2287 

1218 

1260 

+  42 

70 

1599 

852 

884 

+  32 

71 

878 

467 

531 

+  64 

72 

520 

277 

267 

—  10 

73 

262 

139 

118 

—  21 

74 

174 

92 

61 

-  31 

§  137-  SOCIAL   STATISTICS.  I73 

gives  a  comparison  of  the  theoretical  and  observed  heights  of  ! 
18,780  white  soldiers,  including  men  of  all  nativities  and  ages. 
In  the  second  column  are  recorded  the  actual  number  measured 
of  each  height,  and,  in  the  third,  the  proportional  number  in  1 
10,000.  The  mean  height  as  found  by  formula  (9)  is  67.24 
inches,  from  this  the  residuals  are  formed ;  and  the  probable 
error  .'f  a  single  determination,  by  formula  (23),  is  1.676  inches. 
The  theoretical  numbers  between  the  several  limits  are  next 
derived  by  the  help  of  Table  II,  and  recorded  in  the  fourth 
column,  while  the  differences  between  the  calculated  and  ob- 
served numbers  are  given  in  the  last. 

137.  Numerous  comparisons  of  this  kind,  made  by  Quetelet 
and  others,  have  clearly  established  that  stature  and  the  other  , 
proportions  of  the  body  are  governed  by  the  law  of  probability 
of  error.  Nature,  in  fact,  aims  to  produce  certain  mean  pro- 
portions ;  and  the  various  groups  into  which  mankind  may  be 
classified  deviate  from  the  mean  according  to  the  law  of  the 
probability  curve.  And  the  same  is  true  of  intellect.  By  the 
discussion  of  social  statistics,  then,  it  is  possible  to  discovei 
the  mean  type  of  humanity,  not  merely  in  physical  proportion, 
but  in  intellect,  capacity,  judgment,  and  desires.  "The  aver- 
age man,"  says  Quetelet,  "is  for  a  nation  what  the  centre  of 
gravity  is  for  a  body :  to  the  consideration  of  this  are  referred 
all  the  phenomena  of  equilibrium." 

In  fact,  the  distribution  of  social  phenomena  seems  strictly 

analogous  to  that  of  the  rifle-shots  discussed  in  Art.  135.     Each  ' 

shot  may  represent  a  person,  or  some  property  of  a  person,  to  j 

be  investigated.     For  all  the  shots  there  is  a  mean,  showing  ' 

the  most  probable  result ;  and  also,  for  each  group,  there  is  a  ; 
secondary  mean,  depending  on  the  particular  race  or  nation  to 

which  the  person  belongs.     There  is  a  type  for  soldiers,  and  j 

another  for  sailors;  one  for  Americans,  and  another  for  Euro-  ; 

peans  ;  one  for  men,  and  another  for  women  ;  one  for  the  period  1 


174  THE    DISCUSSION   OF  OBSERVATIONS.  IX. 

of  youth,  and  another  for  that  of  maturity.  The  individuals  of 
each  type  are  clustered  around  its  mean,  according  to  the  law 
of  probability  ;  and  the  several  types  are  clustered  around  a 
general  mean,  according  to  the  same  law.  This  is  true  for  all 
statistical  data  in  which  equal  positive  and  negative  deviations 
from  the  mean  are  equally  probable  ;  in  other  cases  an  unsym- 
metric  distribution  may  occur. 

138.   Problems. 

1.  An  angle  is  measured  by  an  instrument  graduated  to  quarter - 
minutes,  the  probable  error  of  a  single  reading  being  12  seconds.  How 
many  observations  are  necessary,  that  it  may  be  a  wager  of  5  to  i  that 
the  mean  is  within  one  second  of  the  truth? 

2.  A  line  is  measured  500  times.  If  the  probable  error  of  each 
observation  is  0.6  centimeters,  how  many  errors  will  be  less  than  i  cen- 
timeter, and  greater  than  0.4  centimeters? 

3.  The  capacity  of  a  certain  large  vessel  is  unknown  :  1,600  persons 
guess  at  the  number  of  gallons  of  water  which  it  will  hold,  and  the 
average  of  their  guesses  is  289  gallons.  The  vessel  is  then  measured 
by  a  committee,  and  found  to  hold  297  gallons.  If  the  probable  error 
of  a  single  guess  be  50  gallons,  and  it  be  impossible  that  there  can  be 
any  constant  source  of  error  in  guessing,  what  is  the  probability  that  the 
committee  have  an  error  in  their  measurement  of  between  3  and  13 
gallons  ? 

4.  Determine  from  the  data  in  Art.  136  the  number  of  men  per 
million  who  are  more  than  seven  feet  tall. 

5.  Two  observations  differ  by  the  amount  a.  A  third  observation 
differs  from  the  mean  of  the  first  two  by  the  amount  «.  Find,  by 
Chauvenet's  criterion,  the  value  of  u  necessary  to  reject  the  third 
observation. 


§  i4^«  THREE  NORMAL   EQUATIONS.  1 75 


CHAPTER   X. 

SOLUTION   OF   NORMAL   EQUATIONS. 

139.  In  the  preceding  pages  the  student  has  been  left  to 
solve  normal  equations  by  any  common  algebraic  process.  It 
is  usual  in  computing  offices,  however,  to  require  them  to  be 
formed  and  solved  by  a  definite  method  for  the  sake  of  uni- 
formity in  making  comparisons.  This  is,  indeed,  absolutely 
necessary  when  the  number  of  unknown  quantities  is  greater 
than  three  or  four,  or  when  the  co-efficients  are  large,  in  order 
that  checks  upon  the  numerical  work  may  be  constantly  had 
and  the  accuracy  of  the  results  be  ensured.  The  methods  in 
most  common  use  will  now  be  explained. 

Three  Normal  Equations. 

140.  The  method  of  elimination,  due  to  Gauss,  which  is  de- 
scribed below,  is  probably  the  best  for  this  case  except  when 
the  co-efficients  are  small  numbers.  In  that  event  the  determi- 
nant formulas  for  solution  may  be  advantageously  employed. 
These  will  be  here  written  for  the  general  case  of  three  linear 
equations, 

A^x  +  Byy  +  C,z  -  Z>„ 
A^x  +  B^y  +  C,z  =  Z>„ 
A^x  +  B^y  -f  Qz  ^  Z?3, 


176  SOLUTION   OF  NORMAL    EQUATIONS. 

the  solution  of  which  gives  the  formulas, 


X. 


X  = 


y  = 


D,  B,   C, 

Z>3  ^3    C3 

A,  D,  C, 

A3,  -^3     ^3 

A,  B,  n, 

A,  B,  A 

^3  ^3  A 


^2  -^2  t^2 

^3  ^^  ^3 

-•^:  ^:  C, 

A  2  Jj  2  *-'2 

.43  ^3  C3 


These  are  readily  kept  in  mind  by  noticing  that  the  denom- 
inator is  the  same  for  each,  and  that  in  the  numerator  the 
absolute  terms  Z>  replace  the  co-efificients  of  the  unknown  quan- 
tity to  be  found.  If  C^  =  €2=  C^  =  o,  and  Aj^=  B^  =  D^  =  o, 
this  solution  reduces  to  that  given  in  Art.  55. 

141.  As  an  illustration  of  this  method  let  the  three  normal 

equations  be 

3JC  -  jK  +  2s  =  5, 

—  jf  +  4>'  +  s  =  6, 
2^  +  J  +  5^  =  3- 
Then  the  determinant  denominator,  being  developed,  gives 


=  32. 


Similarly  the  values  of  the  three  determinant  numerators  are 
found  to  be  no,  86,  and  —  42.     Hence 


3 

—  I      2 

4      I 

—  I     2 

—  I     2 

I 

4      I 

=  3 

1     5 

+  I 

I     5 

+  2 

4     I 

2 

I     5 

^  =  -  fi 


jr   =    4-   A5  1,   _      I      4  3 

•*    —       I       16>  J    —       I       16> 

which  exactly  satisfy  the  three  given  normal  equations. 


§  142.  FORMAl^ION  OF  NORMAL   EQUATIONS.  177 

Checks  upon  the  results  of  the  solution  may  be  also  obtained 
by  writing  the  normal  equations  in  another  order,  making  for 
instance  the  third  the  first  one,  and  thus  obtaining  different 
numerical  determinants  for  development. 

For7natio7i  of  Normal  Eqimtiotis, 

142.  Let  the  n  observation  equations  between  three  unknown 
quantities  be  of  equal  weight,  and  let  the  observed  quantities 
M^,  M„  ,  .  .  M„  be  transposed  to  the  first  term,  giving 

a,x  +  ^,y  +  c^z  +  m,  =  o, 
ajX  +  b^y  +  C2Z  -j-  m^  =  o, 


anx  +  bny  +  CnZ  -{-  m„  =  o, 
and  let  there  be  formed  the  sums 

«i^  +  ^2'  +  .  •  .  +  ^h"  =  [aa]. 


Then  the  three  normal  equations  are 

\aa\x  +  {ab'\y  +  \ac\z  -f  {aTn]  —  o, 
{ba'lx  +  Ibb'Xy  4-  \bc\z  +  [^;//]  =  o, 
\ca\x  +  \cb'\y  +  \cc\z  +  [^/«]  =  o. 

Thus  the  formation  of  the  normal  equations  consists  in  cor.v 
puting  the  co-efficients  \ad],  \ab\  etc.  This  may  be  done  by 
common  arithmetic,  by  the  help  of  Crelle's  multiplication 
ta'ble,  a  logarithmic  table,  a  table  of  squares,  or  a  calculating 
machine,  The  following  method  of  arranging  and  checking 
the  work  is  frequently  employed. 

Write  the  co-efficients  and  absolute  terms  of  the  observation 


178 


SOLUTION   OF  NORMAL    EQUATIONS. 


X 


equations   in    tabular   form  and  add  a  colunnn   containing  the 
algebraic  sums  of  these    for   each   equation.     Thus   for  three 


X 

a 

y 

b 

z 
c 

m 

s 

I 

2 

• 
• 

71 

unknown  quantities  the  table  has  the  above  form,  the  last 
column  containing,  for  each  horizontal  row,  the  algebraic  sum 
a  -\-  d  -{-  c  -\-  m  =  s. 

A  second  table,  which  need  not  be  here  shown,  contains 
fifteen  columns,  headed  aa,  ad,  .  .  .  ss,  and  the  summation  of 
the  products  in  these  columns  gives  the  fifteen  co  ef^cients  and 
absolute  quantities  which  are  arranged  in  a  third  table  as  be- 
low. It  is  to  be  noted  that  [da],  [ca],  [cd]  are  the  same  as  [ad], 
[ac],  [dc],  and  hence  need  not  be  computed. 


X 

}' 

z 

-] 

^1 

Check. 

[a 

[^ 

[c 

[7U 

b 

Here  the  sum  [bb]  is  placed  at  the  right  of  [b  and  under  b],  the 


§143- 


fOKMATJON  OF  NORMAL   EQUATIONS. 


179 


sum  \cs\  at  the  right  of  \c  and  under  s\  and  so  on.  The  last 
column  is  used  to  record  the  results  of  the  five  checks,  namely, 

\aa\  +  \ab\  +  \nc\  +  \ain'\  —  \as\ 
Ibdl  +  \bb\  +  \bc\  +  {bm'\  =  [bs], 
[ca]   +  [cb]  +  [u]   +  [cm]   =  [cs], 
[;//(/]  -j-  [////-]  -f-  [///r]  +  [;/////]  =  \'f/s], 
[sa]    +  [.-^J   +  \sc]    +  [sm]   =  [ss]. 

If  these  checks  are  all  fulfilled,  the  normal  equations  may  be 
regarded  as  coiieclK'  formed.  In  fillin<7  out  the  table  the 
coefficients  [ba].  [w^J,  etc.,  need  not  be  written,  since  they  are 
the  same  as  [crb],  [cm],  etc. 

143.   As  a  simple  example  let  five  observations  upon  three 
quantities  give  the  five  observation  equations 

—  X  -\-  ;:    —     2  =  o, 

—  X     -\-  y  —     gizzo, 

-|-jF  —18=0. 

+  y   —  ^   -    7  =  0, 

-\-  z    —10  =  0. 
The  arrangement  of  the  first  table  is  then  as  follows: 


X 

y 

z 

No. 

a 

b 

c 

7n 

s 

I 

—  I 

0 

+  I 

—      2 

—      2 

2 

—  I 

+  1 

0 

-    9 

-    9 

3 
4 

0 

+  1 

0 

-  18 

-  17 

5 

0 

+ 1 

—   I 

-     7 

-    7 

0 

0 

+  I 

—  10 

-    9 

i8o 


SOLUTION   OF  NORMAL    EQUATIONS. 


X. 


The  products  aa,  ab,  etc.,  are  next  computed,  and  the  sums 
\_aa\  \_ab'\,  etc.,  are  found.  The  table  of  co-efificients  and  ab- 
solute quantities  then  is 


X 

Z 

.;/] 

-3 

Check. 

\a 

+  2 

—  I 

—  I 

+      II 

+    II 

+     II 

\b 

+  3 

—  I 

-     34 

-   zz 

—     IZ 

c 

+  3 

-      5 

—      4 

—       4 

\}n 

+  558 

+  530 

+  530 

\s 

+  504 

-f  5°4 

and  the  checks  being  all  fulfilled  the  computations  are  satis- 
factory.    Thus  the  normal  equations  are 

-\-  2X  —    y  —    2+11=0, 

—  ^  +  3>'  —     ^  —  34  =  o, 

-  -^  -    ;'  +  3=  -    5  =  o. 

and  it  will  be  shown  in  Art.  147  how  these  may  be  solved 
so  as  to  continue  the  above  system  of  checks  throughout  the 
entire  numerical  work. 

The  advantage  of  the  above  system  is  more  apparent  in 
cases  where  the  co-efficients  and  absolute  terms  consist  of 
several  digits  and  where  the  decimals  must  be  rounded  off. 
In  such  cases  the  number  of  decimals  to  be  retained  in  the 
work  should  be  at  least  sufficient  to  cause  the  checks  to  be 
fulfilled  with  an  error  not  greater  than  one  unit  in  the  last 
place.  The  additional  labor  required  for  these  checks  is  fully 
repaid  by  the  assurance  of  correctness  in  the  numerical  work. 


§  144-  GAUSS'S   METHOD    OF  SOLUTION.  l8l 

Gauss  s  Method  of  Solution. 

144.  The  method  of  solution  due  to  Gauss,  by  which  is 
preserved  throughout  the  work  the  symmetry  that  exists  in 
the  coefficients  of  the  normal  equations,  is  extensively  used 
by  computers.  To  illustrate  it,  three  normal  equations  of 
equal  weight  will  be  sufficient. 

From  the  ;/  observation  equations  are  derived,  by  the 
method  of  Art.  142,  the  three  normal  equations 

[aa]a-  +  [<^'^]J^'  +  \ac\z  -\-  [am]  =  o, 
[3a]x  +  [/'d]y  +  [l>c]z  +  [fim]  =  o, 
[ca]x  -|-  [cd]y  +  [cc]::  -\-  [em]  —  o. 

From  the  first  equation  take  the  value  of  x  and  substitute  it 
in  the  second  and  third,  giving 


For  the  sake  of  abbreviation  the  quantities  within  the  paren- 
theses may  be  denoted  by  [dl? .  i],  [dc .  i],  [dpi .  i]  for  the  first 
equation,  and  by  [cd  .  i],  [cc .  1],  [r;;/ .  i]  for  the  second  equa- 
tion.     Then  these  two  equations  may  be  written 

[M .  I  ]  ;•  +  [^c- .  I ]a  +  [hu  .  i]  =  o, 
[cd.i]}'  -}-  [cc .  i]c  +  [cm.  i]  =  o, 

which  are  similar  in  form  to  the  second  and  third  normal  equa- 
tions, except  that  the  terms  containing  x  have  disappeared 
and  each  co-efficient  is  marked  with  a  i.  These  quantities, 
[bb.\\  [bc.\\  may  be  called  "auxiliaries,"  and  the  law  of 
their  formation  is  evident. 


1 82  SOLUTION   OF  NORMAL    EQUATIONS.  X. 

From  the   first  of  these  equations  take  the  value  of  y  and 
substitute  it  in  the  second,  giving 

which  may  be  abbreviated  into 

[cc .  2]s  +  [cm  .  2]  =  o, 

where  [cc.  2]  and  [cm.  2]  may  be  called  "second  auxiliaries." 
The  value  of  the  quantity  ^  now  is 

[cv;/ .  2] 


2  =:   — 


J=   - 


[CC.2]^ 

while  the  values  of  j  and  x  are 

[l>m.  i]      [(5r.  i] 

p^  ~  \jf~rf' 

[«a]       [a«J  [aa]     ' 

and  the  correctness  of  these  results  may  be  tested  by  inserting 
the  computed  values  of  x,  y,  z  in  the  second  and  third  normal 
equations.  Or  the  order  of  computation  may  be  reversed  and 
the  value  of  x  be  first  obtained,  z  being  first  eliminated  and 
then  J ;  this  will  be  necessary  only  in  critical  cases. 

145.  When  the  normal  equations  have  been  formed  by  the 
method  of  Art.  142,  the  checks  there  explained  should  be  con- 
tinued by  the  computation  of  the  auxiliaries  [mi?i .  i],  [bs .  i], 
etc.;  thus, 

And  a  second  table  should  be  formed  for  the  two  equations 
containing  y  and  z,  by  which  four  numerical  checks  are  ob- 
tained. 


§  146.  GAUSS'S  METHOD    OF  SOLUTION.  1 83 

In  the  next  step  also  the  auxiharies   [;«;«,  2],  \cs.2\  etc., 
are  found  ;   for  example, 

and  then  the  third  table  affords  three  numerical  checks. 

146.  A  valuable  final  check  is  obtained  by  computing  the 
third  set  of  auxiliaries  ;  thus, 

lntm.^\^\_mm.2-\ ^^^^^ , 

\7ns .  3 J  =  \_ms .  2 J Ti^^j , 

and  these  three  values  are  equal.  Each  is  also  equal  to  the 
quantity  '^v'^,  or  to  the  sum  of  the  squares  of  the  residuals  ob- 
tained by  substituting  in  the  observation  equations  the  values 
of  X,  y,  and  z,  found  from  the  normal  equations. 

To  prove  this  let  an  observation  equation  be 
ax  -\-  by  -\-  cz  -j-  ;//  =:  o. 

Then  the  most  probable  values,  x,  y,  z,  will  not  reduce  it  to 
zero,  but  leave  a  small  residual  v.     Hence,  strictly, 

ax  -\-  by  -\-  cz  -\-  tn  =  v- 

By  squaring  each  of  the  values  of  i\  and  adding  the  results, 
the  value  of  2v'^  is  found  ;  and  if  from  this  each  normal  equa- 
tion, first  multiplied  by  its  unknown  quantity,  be  subtracted,  iv 
reduces  to 

[am]x  +  [bm]y  +  [cmjz  -}-  [mm]  =  ^z/". 


184  SOLUTION  OF  NORMAL   EQUATIONS.  X. 

If  this  be  regarded  as  a  fourth  normal  equation,  it  becomes, 
after  the  elimination  of  x, 

\bm .  i]j'  +  ^jcin  .  \\z  -\-  \^tn}n .  i]  =^  ^v^^ 

and  after  eliminating  j  it  is 

\cm  .  2]s  -|-  \jnm  .  2]  =  ^v^; 

and  finally,  after  the  elimination  of  z, 

[mm  .  3]  =  -^^'^ 

Hence  the  auxiliary  [/;/;;/ .  3]  is  equal  to  the  sum  of  the 
squares  of  the  residuals  ;  and  that  [//is  .  3]  and  [ss  .  3]  have  the 
same  value  is  shown  by  the  method  of  their  formation. 

147.  As  a  simple  numerical  example  let  the  following  ob- 
servation equations,  all  of  weight  unity,  be  taken: 

—  X  -\-  z  —    2  =  0, 

—  x+j  —     9  =  0, 

+  >'  -  18  =  o, 

+  y  -  ^  -    7  =  0. 

-\-  z  —  10  =  o. 

The  normal  equations  for  this  case  have  already  been  formed  in 
Art.  143,  and  the  values  of  its  co-efficients  and  check  numbers 
will  be  taken  from  the  table  there  given. 

The  computation   of   the  auxiliaries  for  the  two  equations 
containing  J  and  s  is  now  made,  thus: 

■-         -*  "^    -"  [aa\  2 


147- 


GAUSS'S   METHOD    OF  SOLUTION. 


185 


r/  ^         r/    1         {ba^am^  i   X   11 

\l>m.  i]    =  [/H   -        [^'   ^  ~    34  4 ^ —   =  -  28.5, 


[.v.i]     =M 


-33  + 


2 

I    X    II 


27-5. 


\aa\  2 

ycm  .  i]    =  [r..] ^^^^y-  =  -      5  +  -^    =  +  0.5, 

r  T  r     n  [^^Ik-^J  ,     I    X    II  , 

"-  -"  [<ZaJ  2 

r  -1  r  1  ['«a][«wj  ,  o  II    X    II  . 

[;/m  .  I J  =  [wwj p^.—  =  +  558  - -—  =  +  497-5. 

r  n  r         1  [w«][<?^]  ,  II    X    II  .         . 

Ims.  i]    =  [w.]    -  ^^Y    =  +  530  -  -^-  =  +  469-5. 

r      n  T-^^ir^-^l  ,  II    X    II  , 

[^^ .  i]      =  lss\     -    S    V  =  +  504 =  +  443-5. 

and  the  corresponding  tabulation  is  as  follows,  the  four  check.'i 
being  exactly  fulfilled: 


y 

z 

Check. 

3.1] 

+  2.5 

..I] 

-  1-5 

ni  .  I  ] 

..I] 

\p 

-28.5 

-  27.5 

-  27-5 

\c 

+  2.5 

-^0-5 

+  1-5 

+  1-5 

[m 

+  497-5 

+  469-5 

+  469-5 

[s 

+  443-5 

+  443-5 

The  coef^cient  \cc  .2\  and  the  auxiliaries  for  the  final  equa-       | 
tion  in  z  are  next  found  ;  thus, 


1 86  SOLUTION    OF  NORMAL   EQUATIONS. 

[cm  .2]    =  [' 

[cs.2]     =[ 

\mm,  2]  =  [ 

[;«i-  .2]     =  [ 

[ss  .2]      =  [. 
and  the  corresponding  table  with  its  checks  is 


X. 


[cb.ijhs.i]     _ 

''-'^ \UTV^ "'5-0, 

-,        \mb .  i^\bm.  i]         .  . 

[wi^.  iir*^^.  i] .     I    ,z- 


;//w 


\ms 


\ss . 


[w 


2] 


+  1.6 


2] 


-    16.6 
+  172.6 


s .  2] 


-  15-0 
+  156.0 

+  141. o 


Check. 


-     150 
+  156.0 

+  141-0 


The  value  of  the  unknown  quantity  s  now  is 

-  16.6 


1.6 


=  +  10.375. 


and  from  the  two  equations  containing  jj/  and  z, 

^8.5    ,    1-5, 


y  = 


+  — s=  +  17625, 


2.5      2.5 

and  finally,  from  the  first  normal  equation, 

x=  -  V-  +  *-  +  h  =  +  8.500. 
These  values  also  exactly  satisfy  the  second  and  third  normal 
equations. 

Lastly,  the  final  check  of  Art.  146  is  applied  by  computing 
the  third  set  of  auxiliaries  and  the  sum  of  the  squares  of  the 


§148 


WEIGHTED    OBSER  VA  TIONS. 


187 


residuals.  There  are  found  [;//;;/.  3]  =  0.375,  [^-'^•^  •  3]  —  0.375, 
and  [ss  .  3]  =  0.375.  Also,  by  substituting  the  values  of  x,y,  z, 
in  the  observation  equations, 

^'.  =  -0-125,    z',=  -Ko.i25,    e'3— -0.375,    z',=  +  o.25o,    v^^-^o.^it^, 

the  sum  of  whose  squares  is  '2%)^  —  0.375.  Hence  the  correct- 
ness of  all  the  numerical  work  is  assured. 

When    the    coefficients    of    the    normal    equations    contain 
decimals  these  are  to  be  rounded  off  as  the  work  progresses,   ' 
so  that  the  checks  may  be  sufficiently  satisfied. 

Weighted  Observations. 

148.  The  method  of  Gauss  is  also  directly  applicable  to 
normal  equations  derived  from  independent  weighted  observa- 
tion equations.  The  process  will  be  illustrated  for  three 
unknown  quantities.     Let  the  observation  equations  be 


+  X  =  o, 

+y  =  o, 

+  z  =0, 

-\-x—y         +0.92  =  0, 

-y  +  '^+  1-35  =  o, 
—  X         -(-  s  -)-  1. 00  =  o, 

The  first  table  is  then  as  follows  : 


A=    85, 

/>2    =     108, 

A  =    49. 
A  =  165, 

A=    78, 

/e  =     60. 


X 

y 

z 

No. 

/ 

a 

b 

c 

m 

s 

I 

85 

+  1 

0 

0 

0 

+  1 

2 

108 

0 

+  1 

0 

0 

+  1 

3 

49 

0 

0 

+  ^ 

0 

+  1 

4 

165 

+  1 

—   I 

0 

+  0.92 

+  0.92 

5 

78 

0 

—   I 

+ 1 

+  1-35 

+  1-35 

6 

60 

—   I 

—  0 

+  1 

+  1 

+  1 

i88 


SOLUTION   OF  NORMAL  EQUATIONS. 


X. 


Next  the  co-efficients  \^pad\,  \_pab\  etc.,  are  computed  and 
the  table  of  normal  equations  is  formed,  the  co-efficients  below 
the  diagonal  line  being  omitted,  since  [/'<^^?]  is  the  same  as 
\^pab\,  and  so  on. 


X 

y 

2 

Check. 

«] 

^J 

^] 

vi\ 

^] 

\pa 

+  310 

—  165 

-    60 

+  91-8 

+  176.8 

+  176.8 

\pb 

+  35' 

-    78 

-257-t 

—  149.1 

-149.1 

[pc 

+  187 

+  165.3 

+  214-3 

+  214-3 

[pm 

+341-8 

+  34>-8 

+  341.9 

[ps 

+  583-8 

+  583-8 

This  shows  by  its  checks  that  the  computations  are  correct, 
the  discrepancy  between  341.8  and  341.9  being  due  to  the 
rounding  of^  of  decimals.     Thus, 

310X— 165;'—    60s  +    91.8  =  0, 

—  i65.v  +  35iy-    78^-257.1=0, 

-  60A-—    78;'+ 187c  +  165.3  =  0, 

are  the  normal  equations  for  determining  the  most  probable 
values  of  x,  jj/,  and  z. 

149.   The  auxiliaries  S^pbb.\\    [//;r.i],  etc.,  are  computed 
by  exactly  the  same  rules  as  before,  and  the  table  for  the  two 


y 

z 

Check. 

3.1] 

..I] 

7)1  .    l] 

..I] 

Vpb 

+  263.2 

-  109.9 

-    208.2 

-        550 

-549 

Vpc 

+ 175-4 

+    183.  I 

+    248.5 

+  248.6 

^pin 

+   314-5 

{-    289.4 

+  289.4 

\    ^'' 

+   483-0 

+  482.9 

§i5o. 


WEIGHTED    OBSER I  'A  TIONS. 


189 


reduced   normal   equations  containing  y  and   z  is  formed,  the 
four  checks  being  fulfilled  within  one  unit  in  the  last  figure. 

The  second  auxiliaries  \^pcc .2\  [pan.  2],  etc.,  are  computed 
exactly  as   before   and    the   table  for  the  final  equation  in  2  is 


z 

C.2] 

tfl  .  2] 

..2] 

Check. 

[pc 
[p>n 

+    129.5 

+      96.1 
+    149-8 

+   225-5 

+  245-9 
+  471-5 

+  225.6 
+  2459 
+  471-4 

formed  and  its  checks  found   to  be  satisfactory.     The  value  of 
2  now  is 

q6.  I 
2  = ^ =  —  0.7421, 

129-5 
which  is  carried  to  four  decimals  in  order  that^  and  x  may  be 
found  correct  to  two  decimals. 

From   the   first  equations   in   the  two  tables  preceding  the 
last,  the  values  of  jj/  and  x  are  now  obtained,  thus, 

,  208.2    lOQ.g 

310       310       310 

and  hence  ihe  final  results  to  two  decimals  are 

jc  =  —  o.  18,     J  —  ~{-  0.48,     z  =  —  0.74, 
which  are  the  most  probable  values  of  the  unknown  quantities. 

150.    Inserting  these  values  of  x,  j%  s  in  the  six  observation 
equations,  the  residuals  are  found  to  be 

Z'l    =   0.18,       V2   =    +   0.48,       V2,  =    —  0.74,       7'^   =   +  0.26, 

2^5=  +  0-13'     ^'c  =  +  0.44. 


190  SOLUTION   OF  NORMAL   EQUATIONS.  X. 

Squaring  these  and  multiplying  each  square  by  its  correspond- 
ing weight  there  results 

^pv^  =  78.57. 

Ihe  computation  of  the  third  auxiliaries  gives 

[/w;«.3]  ^  78.5,     [/;;/^.3]  =  7S.6,     [pss .  3]  -  -jS.S, 

an  agreement  which  is  as  close  as  is  necessary  for  this  case. 

LogaritJimic  Computations. 

151.  The  use  of  logarithms  is  often  advantageous  in  forming 
the  products  required  in  the  solution  of  normal  equations.  A 
systematic  scheme  for  such  solutions  will  now  be  presented  in 
which  the  four-place  logarithmic  table  given  at  the  end  of  this 
volume  will  be  employed.  In  general  a  five-  or  seven-place 
table  will  be  found  easier  to  use  when  the  co-ef^cients  contain 
more  than  four  significant  figures. 

The  scheme  to  be  used  will  be  as  follows  for  three  normal 
equations,  the  space  for  checks  being  in  a  horizontal  row  at 
the  bottom  and  these  checks  referring  to  the  auxiliaries  instead 
of  to  the  normal  equations  themselves,  which  are  supposed  to 
have  been  first  formed  and  checked  by  the  method  of  Art.  144. 
The  form  is  first  to  be  filled  out  by  writing  the  numbers  \aa\ 
\ab\  .  .  .  \jns\  in  the  places  indicated.  The  logarithms  of  \aa\ 
\ab\  .  .  .  [cis]  are  next  taken  out  and  recorded.  Then  writing 
log  [aa^  on  a  strip  of  paper,  it  is  subtracted  in  turn  from 
log  [nd],  log  [«^j,  log  \ani\  log  \as\  and  the  differences  are 
written,  thus  filling  out  the  top  row  of  squares. 

Log  \ab\  is  now  written  on  a  slip  of  paper  and  added  to  the 
logarithms  at  the  foot  of  the  first  row,  thus  giving  the  loga- 
rithms for  the  second  row.  Those  in  the  third  and  fourth 
rows  are  similarly  found  by  adding  log  \ac\  and  log  \ani\  to 
the   same   ones   as   before.     The    numbers   corresponding  to 


151. 


L  OGAKITHMIC   COMPUTA  TIONS. 


191 


1 

X 

y 

2 

m 

J 

[aa 

log  \aa\ 

[ab 
log  [ab'\ 

\iia\ 

[ac 

log  lac\ 
log  p-^T 

[am 
log  [a/w] 

log   r       T 

log  {as\ 
log  f— T 

[bb 

number 
[bb.\\ 

[be 
number 

[bm 

number 
[^W .  I 

number 
[bS.Y\ 

number 
[cc .  I 

[cm 

number 
[cm.  I 

number 
[ri'.  l] 

[;«wj 

number 
[ww  .  l] 

['iis\ 

number 
[w^ .  I  ] 

Checks. 

'bs.x\ 

Vj.  i] 

[ms .  i] 

[ss .  l] 

[...  I] 

1 

these  logarithms  are  then  taken  from  the  table,  and  each 
number  being  subtracted  from  that  at  the  top  of  the  square, 
the  co-efficients  [bb  i],  [bc.\\,  .  .  .  \jiis .  i]  result.  Lastly  the 
check   \bs .  i]  at  the  foot  of  the  second  column  is  found  by 


192  SOLUTION   OF  NORMAL   EQUATIONS.  X. 

adding  together  \bb .  i],  \bc .  i],  and  \brn .  i]  ;  and  in  a  similar 
manner  [fj-.i]  and  \ins.\\  are  found.  Here  [jt^.i]  may  be 
determined  in  two  ways,  by  the  addition  of  the  horizontal  row 
and  also  by  the  column  above  it. 

A  second  similar  tabulation  is  also  made  for  the  next  opera- 
tion, the  auxiliaries  \bb .  i],  be  .\\  .  .  .  \ins .  i]  being  transferred 
from  the  first  table  to  the  top  of  the  squares  in  the  seconc 
one.  The  process  will  be  now  exemplified  by  a  numerical 
example. 

152.  Let  there  be  given  three  normal  equations  which  have 
arisen  from  a  case  of  conditioned  observations,  namely, 

+  1 7- 73-^  —     4-8q>'  —     8.133  +    4.60  =  o, 

—  4.80X  -\-  1 7. 607  —     2.40s  -|-  34.89  =  o, 

—  8.13JC  —    2.40)'  +  13.933  —    7.75  -  o. 

Here  the  check  sums  [ds],  [bs],  [cs],  [;;/i-]  are  to  be  formed 
from  the  given  co  efficients  ;  for  example, 

[cs]  =  -  8. 13  -  2.40  +  13.93  -  7-  75  =  -  4-35» 

but  [ww],  [w^'],  and  [ss}  cannot  be  obtained.  For  the  purpose 
of  carrying  through  the  full  system  of  checks,  one  of  these, 
say  [w;;/],  may  be  assumed,  and  the  others  be  computed  ; 
assuming  [w/;/]  =  o,  the  value  of  [w.f]  is  -|-  31.74.  The 
co-efficients  and  check  numbers  are  then  arranged  in  the  upper 
right-hand  corners  of  the  squares  in  the  following  table.  The 
four-place  logarithms  of  those  in  the  upper  row  are  taken  out, 
the  letter  n  being  affixed  to  the  logarithm  of  a  negative  num- 
ber. The  subtractions  and  additions  of  these  logarithms  a5, 
required  by  the  scheme  of  the  last  article  are  then  made,  and 
the  corresponding  numbers  taken  from  the  logarithmic  table. 
These  subtracted  from  those  in  the  upper  corners  give  the 
auxiliaries  [bb .  i],  [dc .  i],  etc.,  which  are  written  in  the  lower 


§152. 


LOGARITHMIC   COMPUTA  TIONS. 


193 


right-hand  corners.     The  checks  of  these  are  then  made,  and 
found  to  be  verified  to  one  unit  of  the  last  decimal. 


X 

y 

z 

ni 

s 

+  17-73 

—  4.80 

-8.13 

+  4.60 

+  9.40 

1.2487 

o.68i2« 

o.gioiM 

0.6628 

0.9731 

T.4325« 

i.66i4« 

1-4141 

1-7244 

4-  17.60 

—  2.40 

+  34-89 

+  45-29 

0.1137 

0.3426 

o.og53« 

o.4056« 

+  1.30 

+  2. 20 

-  1-25 

-  2.54 

+  16.30 

—  4.60 

+  36-14 

+  47-83 

+  13-93 

-7-75 

-  4-35 

0.5715 

0.3242// 

o.6345« 

+  3-73 

—  2. 1 1 

-4-31 

+  10.20 

-5-64 

—  0.04 

+  0.00 

+  31-74 

0.0769 

0.3872 

+  1. 19 

+  244 

-1. 19 

+  29.30 

Checks 

+  47-84 

—  0.05 

+  2931 

+  7709 
+  77  10 

The  next  operation  is  to  write  the  values  of  the  auxiliaries 
\bb .  i],  \bc .  i],  .  .  .  \ins .  i]  in  a  second  table  of  squares,  and  by 
a    similar   process    obtain    the    second    set   \cc  .2\,  . .  .\jns  .2\. 


194 


SOLUTION  OF  NORMAL   EQUATIONS. 


X. 


The  scheme  shown  in  the  twelve  upper  left-hand  squares  of 
the  table  in  Art,  151  will  apply  to  this  case  if  a,  b,  c,  in  be 
changed  to  b,  c,  in,  s,  and   i   added  in  all  brackets  except  the 


y 

z 

m 

s 

+  16.30 

—  4.60 

'V  36.14 

+  47-83 

I.2'22 

0.6628M 

1.5580 

1.6797 

T.45o6« 

0.3458 

0.4675 

+  10.20 

-  5-64 

—  0.04 

0.1134 

i.oo36« 

1 . 1 303« 

+  1-30 

—  10.20 

-  1350 

+  8.90 

+  4-56 

+  1346 

-  1.19 

+  29-30 

log  4.56 

=  0.6590 

1.9038 

2.0255 

log  8.90 

=  0.9494 

+  80.13 

+  106.06 

log  z 

=  i.7096« 

-81.32 

-  76.76 

-  63.30 

Check. 

+  13-46 

-  76.76 

-  63.30 

lowest  in  each  square  where  the  I  is  changed  to  2.  The 
operations  are  strictly  analogous  to  those  of  the  preceding 
table. 

A  table  for  the  computation  of  the  third  set  of  auxiliaries 
need  not  be  formed,  these  being  of  no  use,  as  the  sum  [ww] 
was  assumed  at  the  start.     The  value  of  z  now  is 


4.1:6 
s  =  —  ^-^ —      or     2  =  —  log"'  1.7096  =  —  0.512. 
8.90 


§  153-       PROBABLE   ERRORS   OE  ADJUSTED    VALUES.  I95 

From  the  logarithms  in  the  upper  squares  of  the  last  table, 
y——  log"'  (0-3458)  —  log"  (i-45o6«  +  T.7096//)  =  —  2.362, 

and  similarly  from  the   logarithms  in  the  upper  squares  of  the 
first  table,  according  to  the  last  formula  of  Art,  144, 

x=  —  log"'  (T.4141)  —  log-'  (t.66i4«  +t.7096«) 
—  log-' (T.4325//  +  0.3731;/)  =  —  I.  134, 

which   are   the   values   that   closely  satisfy   the    given   normal 
equations. 

After  becoming  acquainted  with  this  method  by  solving 
several  sets  of  normal  equations  the  student  will  find  it, 
except  when  the  coefficients  are  small  integers,  to  be  gener- 
ally more  expeditious  than  methods  which  do  not  employ 
logarithms. 


i3 


Probable  Errors  of  Adjusted  Values. 

153.  When  the  sum  of  the  weighted  squares  of  the  residuals, 
'2pv',  has  been  computed,  the  probable  error  of  an  independent 
observation  of  weight  unity  is  given  by  (32),  namely. 


.=.0.6745/-^^, 
n  —  q 

in  which  n  is  the  number  of  independent  observations  and  q 
the  number  of  unknown  quantities.  If  /^,  /^,  p^  be  the 
weights  of  the  adjusted  values  of  x,  y,  z,  the  probable  errors  of 
these  adjusted  values  then  are 

_     r  _     r  r 

<^-  ypy  ^A 

and  thus  these  are  known  as  soon  as  the  weights  have  been 
determined. 


196  SOLUTION   OF  NORMAL   EQUATIONS.  X. 

154.  To  find  these  weights  the  methods  of  Arts.  74,  75  may 
be  conveniently  employed  for  three  unknown  quantities. 
Using  the  solution  in  Art.  141,  replacing  A^,  B^,  C^,  etc., 
by  [aa],  [<?<^],  [ac'],  etc.,  and  designating  by  D  the  determinant 
denominator  common  to  the  three  values,  there  are  found, 


Z>  D  D 


Py  ~\^.A\rA  _  u.n2'  P'  ~ 


2  • 


^'  ~  [bb^cc]  -  {bcY   ^'      [aalicc]  -  [ac^'   ^'       [aa][bb'\  ~  [ab\ 

which  are  the  weights  of  the  adjusted  values  of  x,  y,  z. 

Referring   again   to  Art.  74,    and  to   the   method  of   Gauss 
given  in  Art.  144,  it  is  seen  that  the  value  of  z  is 

\cm  .2] 


2  =  — 


\cc .  2] ' 


The  negative  sign  here  results  from  the  fact  that  the  absolute 
terms  \atn\  \bt}i\,  etc.,  are  taken  positive  in  the  first  members 
of  the  normal  equations,  and  the  numerator  vanisb.es  when 
those  terms  are  all  zero.  The  quantity  {_cc .  2]  is  thus  tiie 
reciprocal  of  the  co-ef^cient  of  the  absolute  terms  whicli  be- 
longed to  the  normal  equation  for  z  and  is  hence  the  weight 
of  z,  or/j  =  [cc  .  2]. 

]iy  equating  this  value  of /^  to  that  found  above,  D  may  be 
eliminated  from  the  three  expressions,  giving 

.  _  r         1      ^   -  [jLiil'^'Li]      .    -  [^^  ■  2][bb  .  i][aa] 

which  are  values  of  the  weights  expressed  in  terms  of  the 
coefficients  and  auxiliaries  used  in  finding  the  value  of  x,  f,  z, 

155.  For  example,  consider  the  six  observation  equations  of 
Art.  148,  and  let  it  be  required  to  find  the  probable  errors  of 
the  adjusted   values   of  ;r,  y,  z.     The   normal   equations  are 


§  156.       PROBABLE   ERRORS   OF  ADJUSTED    VALUES.  IQ/ 

solved  in  Art.  149,  giving  ^  =  —  o.  1 8,  j  =  -|-  0.48,  z  ^  —  0.74, 
and  the  value  of  '^pv'^  is  found  to  be  78.6  ;  thus, 


^       J  78.6 
r=  0.6745I/  g— —  =  3.45 

is  the  probable  error  of  an  observation  of  weigh;  unity.     The 
weights  of  the  adjusted  values  of  x,  y,  z  are 

12Q.5  X  263.2 

pz  =  1 29- 5.    Py  = -—, =  ^94-4, 

^  75-4 

_  129.5  X  263.2  X  310  _ 

^^  -  '351X187-78^     -  '77-5. 

and  the  probable  errors  of  the  values  of  x,  y,  z  are 

-^c  =  -^^^  =  o.26,     o  = --|^M=  =  0.25,     r.  = -^^14=.  =.  0.30. 
1/177.5  '^"194-4  '^^129, 5 

Accordingly  the  adjusted  values  may  be  written 

a:  =  —  0.18  ±  0.26,    J'  =  +  0.48  ±  o  25,     s  =  —  0.74  ±  0.30, 

which   shows  the   degree   of   mental   confidence  that   the   ad- 
justed values  may  claim. 

156.  When  the  number  of  unknowns  is  large  the  expres- 
sions for  the  weights  of  the  adjusted  values  become  quite 
complex,  and  in  order  to  find  their  values  it  may  be  some- 
times advisable  to  deduce  x,y,  z,  w,  etc.,  by  two  or  more  dif- 
ferent orders  of  elimination.  The  following  are  formulas  for 
the  weights  for  the  case  of  four  unknown  quantities,  where  w 
is  first  determined  and  x  last : 

_  [dd.zWcc    2'][bb.  i_]  [dd.:,'\[cc.2'][bly.iMaa'\ 

^'  ~       \dd .  2],r^^  .1]      '     ^"  \dd .  2Ucr  .  I U^d]~'' 


198  SOLUTION-  OF  NORMAL   EQUATIONS.  "      X. 

in  which  the  subscript  quantities  have  the  following  values, 

IbcY 


\CC  .  l]a  =   \CC\  — 


\bbY 


\dd. 2\  =  [dd]  - \J^Til~         • 

These,  by  omitting  all  factors  containing  d,  reduce  to  the 
same  expressions  as  above  derived  for  the  case  of  three  un- 
knowns, X,  y,  z. 

157.  Problems. 

1.  Three  observations  on  a  single  quantity  furnish  the  observation 
equations  3^  =  2.18,  2x  =  1.44,  4.x  =  2.90.  Find  the  most  probable 
value  of  X  and  its  probable  error. 

2.  Observations  made  in  a  deep  well  near  Paris  on  the  tempera- 
ture at   different  depths  below  the  surface   of  the  earth   gave   the 
following  results,  /  being    the    temperature  corresponding    to   the 
depth  d: 

For  if  =    28  meters,  /  =  11°. 71  C,  for  c/  —  29S  meters,  f  =  22°. 20  C. 

d  =    66  /  =  12.90  d  —  400  /  =  23.75 

^=173  /=  16.40  d  —  S05  /  =  26.43 

d  —  248  t  =  20.00  d  =  54S  f  =  27.70 

Assume  the  temperature  at  the  surface  (^=0),  tobe  the  annual 
mean  /(,  =  10°. 60,  and  also  that  the  law  of  variation  of  /  with  d  is 

given  by 

t  =  to  +Sd+  Td^. 

State  the  observation  equations,  form  the  normal  equations,  solve 
them  by  the  method  of  Art.  154,  and  sliow  that  the  most  probable 
values  of  .5'  and  T  are  4"0'04i53  i  0.00165  and  — 0.00001929  ± 
0.00000356. 


§  157-  PROBLEMS.  199 

3.  Given  the  three  normal  equations, 

6.649X  +  2.041;'  +  2.9413  —  I.OO  =  o, 
2.041X  4-  4.2497+  0.9262  -  1.35  =  o. 
2.941^:  +  0.926)'  +  5.3823  —  0.92  =  o. 

Form  the  sums  \as\  \bs\  \cs\  \^nis\  and  then   solve  the   equations 
by  the  use  of  logarithms. 

4.  At  a  station  O  angles  were  measured  as  follows  between  the 
five  stations,  A,  B,  C,  D,  E : 


AOB  =.     15° 

3/ 

32". 67, 

weight  6, 

AOC  =    45 

20 

47-34, 

weight  4, 

AOD  =  156 

23 

28.76, 

weight  8, 

AOE  -  26S 

44 

19.84, 

weight  3, 

BOC  =    29 

43 

13-56, 

weight  2, 

BOD  =  140 

45 

57-13, 

weight  6, 

BOE  =  253 

06 

45-03, 

weight  I, 

COD  =  111 

02 

42.86, 

weight  4, 

COE  =  223 

23 

30.94, 

weight  8, 

DOE  =  112 

20 

49-32, 

weight  2. 

Let  X,  y,  5,  7C',  be  the  most  probable  corrections  to  the  observed 
values  of  AOB,  AOC,  AOD,  AOE.  State  the  observation  equa- 
tions, form  and  solve  the  normal  equations,  show  that  Z£' =  +o".5i, 
s  =  +  2".36,  etc.,  and  that  the  adjusted  values  of  the  observed 
angles  are  ^(9^  =15°  37'  2>^" .12,  ....  DOE  =  112"  20'  49".i5. 
Also  show  that  the  weight  of  the  adjusted  value  of  AOE  is  8.9  and 
that  its  probable  error  is  ±  o'^.q:. 

5.  Solve  the  following  normal  equations: 

+  380.95x4-142.86^—    16.882  —  68. 637^/ —    36.67=0, 
+  142. 86x  +  428.57^  +    30.962  +    95.00  =  0, 

—  i6.8Sx  +    30.967  +  208.342  —    e.oSif  —  121. 51  =  o, 

—  68.63X  —      6.082  +  8o.547<y  +    16.34  =  O, 

and  show  that  the  value  of  x  is  -|"  0-275. 


200  NON-LINEAR  EQUATIONS.  XI. 


CHAPTER   XI. 

APPENDIX   AND   TABLES. 

158.  The  elementary  principles  and  applications  of  the 
Method  of  Least  Squares  hav^e  now  been  given  and  exempli- 
fied. It  remains  to  note  a  few  points  that  have  not  found  a 
place  in  the  preceding  chapters,  to  present  some  remarks  on 
the  history  and  literature  of  the  subject,  and  to  give  several 
tables  that  will  be  useful  in  abridging  computations. 

Observations  Involving  Non-Linear  Equations. 

159.  It  has  been  thus  far  assumed  that  the  observations  can 
be  represented  by  equations  of  the  first  degree.  If  this  is  not 
the  case,  and  higher  equations  are  involved,  they  can  be  re- 
duced to  linear  ones  by  the  following  method  : 

Let  the  q  quantities  to  be  determined  be  represented  by 
;?,,  z^  .  .  .  Zg,  a>nd  the  n  measured  quantities  by  J/„  M2  .  .  .  M„, 
and  let  the  n  observation  equations  be  of  the  form 

^  =/(So  Z2  ..  .  2?)  =  M. 

Now,  let  approximate  values  of  the  unknown  quantities  be 
found,  either  by  trial,  or  by  the  solution  of  a  sufficient  number 
of  equations,  and  let  them  be  denoted  by  if,,  Z^  .  .  .  Zg.     Let 


§  159-  APPENDIX  AND    TABLES.  20T 

3,,  z^  .  .  .  z'  be  the  most  probable  corrections  to  these  ap- 
proximate values  ;  so  that 

If,  now,  each  of  the  functions  <^  be  developed  by  Taylor's 
theorem,  and  the  products  and  higher  powers  of  the  correc- 
tions be  neglected,  there  will  be  7i  expressions  of  the  form 

<^  =/(Z.,  Z3  .  .  .  Z,)  -M+'±z:^  ^.;  +  . . .  +  -^,^'  =  o. 

dz,  dz^  dZg  ^ 

Here  the  terms  /(Z„  Z,  .  .  .  Z,)  are  known,  and  may  be  desio-. 
nated  by  N,,  N,  .  .  .  N,;  so  that  the  n  expressions  reduce  t'^o 
the  form 

fv  +  jA'+...  +  ^v  =  ^/-^, 

where  the  differential  co-efificients  are  to  be  found  by  differen- 
tiating each  of  the  observation  equations  with  reference  to 
each  variable,  and  then  substituting  the  approximate  values 
Z„  Z^  .  .  .  Zg,  for  z^,  Z2  .  .  .  Zg.  Denoting  them,  then,  by  a,  b,  c, 
etc.,  the  71  equations  are  of  the  form 

az,  +  dz^  +  cz^  +  ..  .  +  Izq  =  M-  N, 

in  which  all  the  letters  except  z„  z,  .  .  .  z„  denote  known 
quantities.  These  ;/  equations  are  exactly  like  the  observa- 
tion equations  (10)  or  (12),  and  from  them  the  normal  equa- 
tions are  formed,  whose  solution  furnishes  the  most  probable 
values  of  the  corrections. 

If  non-linear  conditional  equations  are  given,  it  is  also  neces- 
sary to  find  approximate  values  for  the  unknown  quantities, 
and  assume  a  system  of  corrections.  Then  the  functional  con- 
ditional equations  may  be  developed  as  above,  and  reduced  to 


202  NON-LINEAR   EQUATIONS.  XI. 

linear  equations  containing  the  corrections  as  unknowns,  which 
may  be  treated  by  the  method  of  correlatives,  and  the  most 
probable  system  of  corrections  determined,  which,  applied  to 
the  approximate  values,  will  give  the  adjusted  results.  If  these 
do  not  satisfy  the  original  equations  with  sufficient  accuracy,  a 
new  system  of  corrections  should  be  assumed,  and  the  process 
be  again  repeated. 

Certain  expressions,  like  that  in  Art.  in,  may  be  reduced  to 
the  linear  form  by  the  help  of  logarithms  ;  and,  when  this  is 
possible,  it  will  be  found  a  more  convenient  method  of  treat- 
ment than  the  development  by  Taylor's  theorem. 

l6o.  As  an  illustration  of  the  method,  let  M  be  the  numbei 
of  millions  of  people  under  the  age  of  m  years,  and  let  it  be 
required  to  find  the  most  probable  value  of  z  in  the  empirical 
formula 

</)  =  50.16  sin  ;« (0.996)  "'2  =  M, 

which  is  supposed  to  give  the  relation  between  M  and  m  for 

the  population  of  the    United   States  in   1880.     The  data  are 

nine  values  of  M,  from  the  census  compendium,  given  in  the 

second  column  of  the  table  below. 

The  first  step  is  to  find  by  trial  that  i°.55  is  an  approximate 

value  of  the  angle  z.     The  second  is  to  compute  nine  values  of 

the  expression 

50.16  sin  w  (0.996)'"!°. 5 5  =  N, 

corresponding  to  the  nine  given  values  of  ni :  these  are  put  in 
the  third  column  of  the  table.  In  the  fourth  column  are  the 
differences  M  —  N  between  the  observed  and  computed  values. 
The  fifth  column  contains  the  values  of  the  derivative 

—  =  5o.i6w(o.996)'"cos  ;«(o.996)'"i°.ss, 
dz 

corresponding  to  the  nine  values  of  m. 


§  l6o. 


APPENDIX  AND    TABLES. 


203 


m. 

M. 

N. 

M—  N. 

dz 

10 

13-39 

12.90 

+  0.49 

466 

20 

24.1  2 

24.06 

-f-  0.06 

814 

30 

33-29 

33-07 

+  0.22 

1004 

40 

39.66 

40.08 

—  0.42 

1032 

50 

44.22 

44.98 

—  0.76 

913 

60 

47-33 

48.10 

-  0.77 

674 

70 

49.16 

49-74 

-0.58 

349 

80 

49.94 

50.14 

—  0.20 

29 

90 

50.14 

49.67 

+  0.47 

447 

Let  z'  be  the  most  probable  correction  to  the  assumed  value 
of  z.  Then  the  last  two  columns  furnish  the  nine  observation 
equations 

466s'  =  -f-  0.49, 

8142'  =  +  0.06,     etc. 

From  these  the  single  normal  equation  is  formed,  and  its  solu- 
tion gives 

2'  =  —  0^^.00025  =  —  o°.oi4, 

and  hence  the  most  probable  value  of  z  is 

2  =  i°.55  -  o°.oi4  =  i°.S36; 

so  that  the  empirical  formula  may  be  written 

J/=  50.i6sinw(o.996)'"i°.536. 

When  2"  is  i°.55,  the  sum  of  the  squares  of  the  residuals  is 
about  2.25  ■;  and,  when  .s-  is  i°.536,  that  sum  is  about  1.67 ;  so  that 


204 


MEAN  ERROR   AND   PROBABLE   ERROR.  XI. 


the  precision  of  the  formula  has  not  been  greatly  increased  by 
the  variation  in  the  angle  z.  By  slightly  increasing  the  number 
0.996,  the  formula  can  be  made  to  more  closely  agree  with  the 
observations. 

Mean  Error  and  Probable  Error. 

161.  The  probable  error  is  an  error  of  such  a  value  that  any 
given  error  is  as  likely  to  exceed  it  as  be  less  than  it,  and  it 
hence  seems  to  be  the  quantity  that  would  most  naturally  be 
selected  for  indicating  the  precision  of  observations.  But  there 
is  another  error  very  commonly  employed  for  the  same  purpose 
called  the  "mean  error,"  whose  definition  is,  the  error  whose 
square  is  the  mean  of  the  squares  of  all  the  errors.  Hence 
the  mean  error  is,  for  direct  observations,  the  square  root  of  the 

V    ..2 

quantity  ^— ,  or,  in  terms  of  the  residuals,  the  square  root  of 
n 

the  quantity  ■ — ''- — .     In  general,  then,  the  mean  error  can  be 

n  —  I 
determined  from  the  formulas  for  probable  error  by  changing 

the  co-efficient  0.6745  into  unity.     If  ni  be  the  mean,  and  r  the 

probable  error,  the  relation  between  them  is 

7' 

m  =  =  1.4826^. 

0.6745 

In  the  annexed  figure,  OP  indicates  the  probable  error,  and 
OM  the  mean  error.  It  is  seen,  by  Art.  29,  that  M  is  the 
abscissa  of  the  point  of  inflection  of  the  probability  curve. 

In  Table  II,  the  value  of  the  integral  for  the  argument 
1.4826;'  is  0.6826.  Hence  0.6826  is  the  probability  that  an 
error  is  less  than  the  mean  error,  or  in  1,000  errors  there 
should  be  683  less  than  m.  It  is  a  fair  wager  of  i  to  i  that 
an  error  taken  at  random  is  less  than  the  probable  error;  but  it 
is  a  fair  wager  of  683  to  317,  or  about  2.15  to  I,  that  it  is  less 
than  the  mean  error. 


§   !62. 


APPENDIX  AND    TABLES. 


205 


The  mean  error  is  generally  used  in  German  books.  In  this 
country  the  probable  error  is  commonly  employed  ;  and,  being 
the  most  natural  unit  of  comparison,  it  is  certainly  to  be  desired 
that  it  alone  should  be  used,  and  the  mean  error  be  discarded. 


M     P  O  P     M  U 

Fig.  14. 

162.  Instead  of  the  mean  or  probable  error,  a  quantity 
called  the  "huge  error"  might  be  employed  to  indicate  the 
precision  of  measurements.  The  huge  error  is  defined  to  be 
an  error  of  such  a  magnitude  that  999  errors  out  of  1,000  are 
less  than  it,  and  only  i  greater  ;  or,  in  other  words,  that  the 
probability  of  an  error  being  less  than  it,  is  0.999.  ^^  ^^  be 
the  huge  error,  the  relation  of  ;/  to  r  is  found  from  Table  II. 

For  P  =.  0.999,  the  argument      is  4.9  :  hence 

r 

ti  —  4.gr. 

Accordingly,  all  formulas  for  probable  error  may  be  changed 
into  those  for  huge  error  by  writing  t,.^  in  place  of  0.6745. 
For  instance,  the  huge  error  of  a  single  direct  observation  is 
given  by 


u 


V  «  —  I 


In  Fig.  14  the  abscissa  OU  represents  the  huge  error,  and  the 
area  UDADU  \s  0.999  of  the  total  area. 


206  APPENDIX  AND    TABLES.  XI. 


Uncertainty  of  the  Probable  Error, 

163.  The  value  of  the  probable  error  r,  deduced  in  Art.  67, 
is  the  best  attainable  value,  or  rather  the  most  probable  value. 
The  inquiry  is  now  to  be  made  as  to  what  are  the  probable 
limits  of  this  value  of  r,  or  what  is  the  probable  error  of  the 
probable  error.     Or,  if  the  probable  error  ;-  be  written  in  the 

form 

r{i  ±  u), 

the  number  u  is  the  uncertainty  of  the  probable  error,  that  is, 
it  as  likely  that  the  value  formed  for  r  lies  between  the  limits 
;-(i  —  u)  and  r(i  -\-  n)  as  that  it  lies  outside  those  limits. 
Thus  iir  may  properly  be  called  thw  probable  error  of  the 
probable  error  r. 

164.  A  series  of  observations  having  been  made,  all  having 
the  same  measure  of  precision  h,  the  sum  of  the  squares  of  the 
errors  is  a  constant,  while  the  probability  of  any  value  h 
is,  by  Art.  65, 

P'  ■-  /i"{dxy'7T  -  i"e  -  ^^'2-^% 

and  the  value  of  /i  which  renders  this  a  maximum  is  the  most 
probable  value  of  /i.  Now  let  //  -j-  n/i  be  a  value  greater  than 
this  probable  value  ;  then 

/"'  =  (//  4-  u/i)"{£x')"7r  -  i"e  -  (^'  +  "^""2-^" 

is  the  probability  of  the  value  /i  -\-  nil.  The  ratio  of  these 
probabilities  is 

pn 
P' 


_=  (l  ^^^)v-(2«  +  «=)A»2^='^ 


and  taking  the  logarithms  of  both  sides  of  this  equation, 

P" 

log  —,  =  «  log  (i  +  «)  —  {211  -\-  u')h^'2x'\ 


§  164.        UNCERTAINTY   OF   THE   PROBABLE   ERROR.  20/ 

also    replacing    log  (i  -\-  u)  by   «  —  ^?^^   the    terms    involving 
higher  powers  of  u  being  omitted,  there  results 

F" 

log  —^--[n  —  2h':2x')ii  —  {U  +  /i^2x^)u\ 

The  value  of  h  deduced  in  Art.  65  causes  the  co-efificient  of  it 
to  become  zero,  whence 

log  -^  =  -  nil-     and     -pf  =  ^"""  • 

Thus  the  probability  of  the  variation  uh  in  the  value  of  h  is 
expressed  by  the  function 

P"  =  ce-""\ 

which  is  of  the  same  form  as  the  law  of  accidental  error. 

The   probability   that  71  is  less  than   any  assigned   limit   is 
therefore,  as  in  Art.  32,  expressed  by  the  integral 


V 
and  the  value  of  this  integral  is  |-,  as  in  Art.  61,  when 

t  ^=  u  Vn  =  0.4769. 
Consequently  the  probable  error  of  the  measure  of  precision  /i  is 

0-4769 

u  =  — -^-, 

and  hence  the  probable  limits  of  /i  are 

Thus  the  uncertainty  in  the  probable  value  of  /i  has  been  found. 
Now,  since  Ar  =  0.4769,  the  uncertainty  in  the  value  of  r  is 


—      I> 


208  APPENDIX  AND    TABLES.  XI. 

the  same  as  that  in  the  value  of  Ji.  The  probable  limits  of  the 
probable  value  of  the  probable  error  r  are,  therefore, 

/         0.4769  \           ,        (      ,     0-4769 
r\  I ^^         and     r\\  -\-  ^ 

and  the  uncertainty  in  r  decreases  directly  as  the  square  root 
of  the  number  of  observations.  Thus  for  four  observations 
the  uncertainty  in  r  is  24  per  cent  of  its  value,  for  16  observa- 
tions it  is  12  per  cent  of  its  value. 

165.  The  above  supposes  that  the  probable  error  is  computed 
by  the  sums  of  the  squares  of  the  residuals  according  to  for- 
mulas (20)  and  (21).  If,  however,  formulas  (35)  and  (36)  be 
employed,  using  the  sum  of  the  residuals  only,  then  a  similar 
investigation  will  show  that 

i,_J^i)      and     ^(.  + -°-52|5l 

are  the  probable  limits  of  the  probable  error  r.  Here  the 
uncertainty  is  greater  than  in  the  former  case,  1 14  observations 
being  necessary  to  give  the  same  uncertainty  in  the  probable 
error  as  lOO  observations  give  when  (20)  and  (21)  are  used. 

It  may  be  noted,  finally,  that  some  writers  state  the  above 
expressiop.s  for  tlie  uncertainty  so  that  Vn  —  i  appears  in  the 
denominator  instead  of  V tt. 

The  Median. 

166.  When  an  odd  number  of  direct  measurements  are  made 
on  a  single  quantity,  the  middle  one  in  the  order  of  numerical 
magnitude  is  called  the  median.  Thus,  if  the  results  of  nine 
direct  observations  are 

103,        104,       105,        106,       106,       107,       108,       no.       III, 

the  fifth  one,  counting  from  either  end,  is  106,  which  is  the 
median. 


§  J  6;.  XHE   MEDIAN. 


209 


If  the  number  of  observations  be  even,  the  median  is  the 
mean  of  the  two  middle  ones  in  the  order  of  magnitude. 
Thus,  if  to  tlie  above  observations  there  be  added  112,  then 
the  median  is  |(io6 -f  107)  =  io6|.  In  the  first  case  the 
arithmetical  mean  is  io6|  and  in  the  second  case  it  is  107.2. 
The  median  in  general  difYcrs  from  the  arithmetical  mean. 

When  observations  are  weighted  these  weights  are  to  be 
i.'sed  in  counting  off  the  large  and  small  observations  until  the 
middle  one  or  the  two  middle  ones  are  found,  and  then  an 
interpolation  is  made  to  find  the  median.      For  example,  let 

Observation  =  i,     2,        3,       4,     5, 
Weight  =  2,     5,     16,      10,     7. 

Here  the  sum  of  the  weights  is  40,  which  may  be  taken  as  the 
total  number  of  direct  observations,  and  the  median  plainly  lies 
between  2\  and  3^.  Seven  observations  are  less  than  2\  and 
seventeen  are  greater  than  i\\  thus  sixteen  observations  may 
be  said  to  lie  between  2\  and  3^^,  and  this  interval  is  to  be 
divided  in  the  ratio  of  20  —  7  to  20  —  17.  The  median  hence  is 
2i  +  if  =  3tV  «•■  again  3i  -  -3^  =  3-j^v. 

167.  The  probable  error  of  a  single  observation  is  to  be 
found  by  counting  off  one-fourth  of  the  residual  errors  from 
both  ends,  and  if  these  are  not  equal  their  mean  may  be 
taken.     Thus,  for  the  following  case  wheie  the  median  :s  33, 

Observation  =  31,     32,     32,     i^,     33,     34,     35,     36, 
Residual         =     2,       i,       i,       o,       o,        i,       2,       3, 

the  probable  error  found  by  counting  off  two  residuals  from 
the  left  is  i.o,  while  by  counting  ofT  two  from  the  right  it  is 
15,  the  mean  of  these  being  1.25,  and  then 


r. 


-  I:^  - 


^  =  0-44, 


is  the  probable  error  of  the  median  itself. 


2IO  APPENDIX  AND    TABLES.  XI. 

The  median  was  first  suggested  by  Galton  in  1875*  as  a  con- 
venient method  of  obtaining  a  mean  without  the  necessity  of 
making  man]'  measurements.  For  example,  if  it  were  desired 
to  obtain  the  mean  height  of  the  boys  in  a  school  they  might 
be  arranged  in  a  row  in  the  order  of  height  and  then  the 
measurement  of  the  middle  boy  would  give  the  median. 
Further,  if  the  probable  variation  in  height  were  required  it 
would  be  only  necessary  to  measure  the  two  boys  standing  at 
the  quarter  points  of  the  line,  and  then  subtract  the  mean  of 
their  heights  from  the  median.  This  gives  the  probable  error 
of  a  single  height,  and  by  dividing  it  by  the  square  root  of  the 
number  of  boys  the  probable  error  of  the  median  height  is 
obtained. 

The  median,  when  obtained  by  the  process  indicated  by 
Galton,  may  be  regarded  as  a  representative  value  of  tiie 
mean  quantity  which  is  desired.  But  when  all  the  individual 
measures  are  actually  taken,  the  arithmetical  mean  and  not 
the  median  is  the  most  probable  value,  provided  that  the  law 
of  variation  is  the  same  as  the  law  of  facility  of  accidental 
error.  To  take  the  median  in  the  latter  case,  for  the  sake  of 
avoiding  computation,  can  only  be  justified  when  the  observa- 
tions are  rough  ones,  and  then  the  median  itself  is  liable  to 
differ  considerably  from  the  arithmetical  mean.  The  use  of 
the  median,  except  in  the  manner  indicated  by  Galton,  does 
not  seem  warranted  in  cases  of  symmetric  probability. 

The  uncertainty  of  the  probable  error  of  the  median  is 
greater  than  that  of  the  arithmetical  mean,  217  observations 
being  necessary  in  the  former  case  to  give  the  same  uncer- 
tainty as  100  observations  give  in  the  latter  case.f 


*  Statistics  by  Intercomparison,  Philosophical  Magazine,  vol.  xlix,  p.  33. 

f  See  Gauss,  Werke,  vol.  iv,  pp.  109-117.  See  also  Scripture,  On  mean 
values  from  direct  measurements,  in  Studies  from  Yale  Psychological  Labora- 
tory, 1S94,  vol.  ii,  pp.  1-39. 


§168. 


HISTOR  V  AND  LITER  A  TURE.  2 1 1 


History  and  Literature. 


i68.  The  average  or  arithmetical  mean  has,  from  the  earhest 
times,  been  employed  for  the  determination  of  the  most  proba- 
ble value  of  a  quantity  observed  several  times  with  equal  care. 
From  this  arises  so  naturally  the  idea  of  weights  and  of  the 
weighted  mean,  that  undoubtedly  both  were  in  use  long  before 
any  attempt  was  made  to  deduce  general  laws  based  upon 
mathematical  principles.  About  the  year  1750  certain  indi- 
rect observations  in  astronomy  led  to  observation  equations, 
and  the  question  as  to  the  proper  manner  of  their  solution 
arose.  Boscovich  in  Italy,  Mayer  and  Lambert  in  Germany, 
Laplace  in  France,  Euler  in  Russia,  and  Simpson  in  England, 
proposed  different  methods  for  the  solution  of  such  cases,  dis- 
cussed the  reasons  for  the  arithmetical  mean,  and  endeavored 
to  determine  the  law  of  facility  of  error.  Simpson,  in  1757,  was 
the  first  to  state  the  axiom  that  positive  and  negative  errors 
are  equally  probable;  and  Laplace,  in  1774,  was  the  first  to 
apply  the  principles  of  probability  to  the  discussion  of  errors 
of  observations.  Laplace's  method  for  finding  the  values  of  q 
unknown  quantities  from  n  observation  equations  consisted  in 
imposing  the  conditions  that  the  algebraic  sum  of  the  residuals 
should  be  zero,  and  that  their  sum,  all  taken  with  the  positive 
sign,  should  be  a  minimum.  By  introducing  these  conditions, 
he  was  able  to  reduce  the  ;/  equations  to  q,  from  which  the  q 
unknowns  were  determined.  This  method  he  applied  to  the 
deduction  of  the  shape  of  the  earth  from  measurements  of  arcs 
of  meridians,  and  also  from  pendulum  observations. 

The  honor  of  the  first  statement  of  the  principle  of  Least 
Squares  is  due  to  Legendre,  who  in  1805  proposed  it  as  an 
advantageous  and  convenient  method  of  adjusting  observations. 
He  called  it  "  Methode  des  moindres  quarres,"  showed  that 
the  rule  of  the  arithmetical  mean  is  a  particular  case  of  the 


212 


APPENDIX  AND    TABLES.  XI. 


general  principle,  deduced  the  method  of  normal  equations,  and 
gave  examples  of  its  application  to  the  determination  of  the 
orbit  of  a  comet  and  to  the  form  of  a  meridian  section  of 
the  earth.  Although  Legendre  gave  no  demonstration  that 
the  results  thus  determined  were  the  most  probable  or  best, 
yet  his  remarks  indicated  that  he  recognized  the  advantages  of 
the  method  in  equilibrating  the  errors. 

The  first  deduction  of  the  law  of  probability  of  error  was 
given  in  1808  by  Adrain,  in  "The  Analyst,"  a  periodical  pub- 
lished by  him  at  Philadelphia.  From  this  law  he  showed  that 
the  arithmetical  mean  followed,  and  that  the  most  probable 
position  of  an  observed  point  in  space  is  the  centre  of  gravity 
of  all  the  given  points.  He  also  applied  it  to  the  discussion  of 
two  practical  problems  in  surveying  and  navigation. 

In  1809  Gauss  deduced  the  law  of  probability  of  error  as  in 
Arts  27,  28,  and  from  it  gave  a  full  development  of  the  method. 
To  Gauss  is  due  the  algorithm  of  the  method,  the  determi- 
nation of  weights  from  normal  equations,  the  investigation  of 
the  precision  of  results,  the  method  of  correlatives  for  condi- 
tional observations,  and  numerous  practical  applications.  F"ew 
branches  of  science  owe  so  large  a  proportion  of  subject-matter 
to  the  labors  of  one  man. 

The  method  thus  thoroughly  established  spread  among  as- 
tronomers with  rapidity.  The  theory  was  subjected  during  the 
following  fifty  years  to  rigid  analysis  by  Encke,  Gauss,  Hagen, 
Ivory,  and  Laplace,  while  the  labors  of  Bessel,  Gerling,  Hansen, 
and  Puissant,  developed  its  practical  applications  to  astronom- 
ical and  geodetical  observations.  During  the  period  since  1850, 
the  literature  of  the  subject  has  been  greatly  extended.  The 
writings  of  Airy  and  De  Morgan  in  England,  of  Liagre  and 
Quetelet  in  Belgium,  of  Bienayme  in  France,  of  Schiaparelli 
in  Italy,  of  Andra  in  Denmark,  of  Helmert  and  Jordan  m 
Germany,  of  Chauvenet  and  Schott  in  the  United  States,  have 


§  169.  HISTORY  AND   LITERATURE.  21  3 

brought  the  science  to  a  high  degree  of  perfection  in  all  its 
branches,  and  have  caused  it  to  be  universally  adopted  by  scien- 
tific men  as  the  only  proper  method  for  the  discussion  of 
observations. 

169.  In  1877  the  author  published,  in  the  "Transactions  of 
the  Connecticut  Academy,"  a  list  of  writings  relating  to  the 
Method  of  Least  Squares  and  the  theory  of  the  accidental 
errors  of  observation,  which  comprised  408  titles.  These  were 
classified  as  313  memoirs,  72  books,  and  23  parts  of  books. 
They  were  written  by  193  authors,  127  of  whom  produced  only 
one  book  or  paper  each.  The  date  of  publication  of  the  earliest 
is  1722.  From  that  time  to  1805,  the  year  of  Legendre's  an- 
nouncement of  the  principle  of  Least  Squares,  there  are  22 
titles  ;  since  1805  there  is  a  continual  yearly  increase  in  the 
number ;  thus  : 

From  1805  to  1S14  inclusive,  there  are  18  titles. 

From  1815  to  1824  inclusive,  there  are  30  titles. 

From  1825  to  1834  inclusive,  there  are  32  titles. 

From  1835  to  1844  inclusive,  there  are  45  titles. 

From  1845  to  1854  inclusive,  there  are  63  titles. 

From  1855  to  1864  inclusive,  there  are  71  tides. 

From  1865  to  1874  inclusive,  there  are  95  titles. 

The  books  and  memoirs  are  in  eight  languages  ;  and,  classified 
according  to  the  place  of  publication,  they  fall  under  twelve 
countries.  It  may  be  interesting  to  note  the  number  belonging 
to  each  ;  thus  : 


Countries. 

Germany 153 

France 78 

Great  Britain     .     .     .     .  56 

United  States    ....  34 

Belgium 19 

Russia 16 

Italy 14 


Countries. 

Austria 10 

Switzerland 9 

Holland 7 

Sweden    ......  7 

Denmark 5 

Total 408 


214 


APPENDIX  AND    TABLES. 


XI. 


Languages. 

German 167 

French no 

English 90 

Latin 16 

Itahan 9 


Languages. 

Dutch 7 

Danish 5 

Swedish 4 

Total 408 


The  titles  of  papers  and  books  issued  since  1876  maybe  mostly 
found  in  the  excellent  publication  "  Jahrbuch  iiber  die  Fort 
schritte  der  Mathematik."* 


Constant  Numbers. 

170.  In  the  preceding  pages  the  constant  numbers  entering 
the  formulas  for  probable  error  have  been  stated  only  to  four 
decimal  places,  which  is  entirely  sufficient  for  any  practical 
computation.  As  a  matter  of  mathematical  interest,  however, 
they  are  here  given  to  seven  decimals,  together  with  a  few  other 
related  constants  and  their  common  logarithms. 


Symbol. 

Constant. 

Logarithm. 

hr 

0.4769363 

T.6784604 

hr\j2 

0.6744897 

1.8289754 

hr^Tz 

0-8453476 

1-9270353 

S/2 

I.4142136 

O.1505150 

77 

3-1415927 

0.4971499 

V/tt 

1-7724539 

0.2485749 

TT-I 

0.5641896 

1-7514251 

e 

2.7182818 

0.4342945 

Mod. 

0-4342945 

T.6377843 

*  Gore's  Bibliography  of  Geodesy,  published  in  the  U.  S.  Coast  and  Geodetic 
Survey  Report  for  1887,  will  be  found  excellent  on  the  subject  of  the  method  of 
least  squares. 


§175.  ANSWERS    TO  PROBLEMS;    AXD  NOTES.  275 

Answers  to  Problems  ;  and  Notes. 

171.  Below  are  given  answers  to  a  number  of  the  problems 
stated  in  the  text  and  hints  concerning  the  solution  of  others, 
together  with  explanatory  notes  upon  some  of  the  more  diffi- 
cult  points  in  the  theory  of  the  subject. 

Article  16. — Problem  2:  ^.  Problem  3:  0.9308  by  the 
use  01  /able  V.  Problem  5  :  Fmd  the  probability  of  a  hun- 
dred heads  in  a  single  throw  of  a  hundred  coins,  and  multiply 
this  by  the  number  of  inhabitants  and  the  number  of  seconds 
to  find  the  probability  of  the  occurrence  under  the  given  data. 
Problem  6:  The  probabihty  that  the  nickel  is  in  the  first  purse 


IS  ^ 


1  9- 


Article  26. — The  equation  at  the  foot  of  page  20  may  be 
written  in  the  form 

y  —  y'  __      2{/lx  —  x) 


y  {m  -{-  2)/ix  —  x^ 

and,  in  passing  to  the  limit,  y  -~  y'  is  infinitely  small  compared 
to  J,  and  z/,r  vanishes  with  respect  to  x.  Hence  in  the  second 
member  2x  is  infinitely  small  compared  to  the  denominator, 
and  accordingly  x  vanishes  with  respect  to  {in  -{-  2)Ax. 

Article  37. — Problem  2  :  see  Fig.  2  and  Fig.  6.  Problem  3  : 
Show  this  by  the  principle  of  sufficient  reason.  Problem  5  : 
because  k  depends  upon  h  and  h  is  difTerent  in  the  two  cases, 
Problem  6:  P^rom  formula  (2)  an  expression  for  n^  is  found, 
then  //,  dx,  x,  and  y  are  derived  by  observation  and  tt  is  com- 
puted ;  thus  for  the  case  of  Article  ^t,  the  probability  of  the 
error  3". 5  may  be  roughly  taken  as  that  of  the  occurrence 
between  the  limits  3''.o  and  4" .0,  so  that  the  observed  value  oi 
y  is  Y^ y,  and  as  dx  is  i''.o,  there  results 


.  /idx  100    X    I 


yg/i-^J:'^  242.236    X    11-57 


2l6  APPENDIX  AND    TABLES.  XI. 

whence  n  ■=  4.48,  a  rude  result  indeed,  but  by  increasing  the 
number  of  observations  and  decreasing  ihj  interval  between 
the  successive  errors  a  closer  accordance  may  be  secured. 

Article  59. — Problem  2 :  N.  2 '.4  E.  Problem  3 :  ^,  — 
—  0.19,  ^2  =  +0-14.  ^3  =  +0-05,  etc.  Problem  4:  -^^d  to  A^ 
^^  to  B,  and  i|  to  C,  the  greater  the  weight  the  less  ^?.ing  the 
amount  of  correction. 

Article  6j. — The   reason  why    '^px^y  is  the  same  as ■ 

is  sometimes  not  clear  to  students.  If  each  term  such  as /,;»;,* 
occurs  ny^  times  in  n  observations,  then 

P,x,\n)\  -\- P-,x^-.ny^  +  etc.  =  '^px', 
or 

n{p^x^)\  +/2^o)'s  +  etc.)  =  '2px^ ; 

whence,  dividing  by  n,  follows  the  statement  as  given. 

Article  89. — Problem  i:  o".4o8.  Problems  3  and  4:  The 
combination  of  observations  differing  widely  in  precision,  as 
in  these  examples,  is  not  ahva}'s  safe  in  practice,  because  of 
the  constant  errors  which  are  liable  to  affect  the  less  precise 
series,  so  that  the  practical  weight  of  the  more  precise  series  is 
often  greater  than  that  derived  from  the  probable  errors. 
Problem  6:  It  should  be  inferred  that  a  constant  source  of 
error  exists. 

Article  98. — Pioblem  2  :  0,000137,  which  occurs  when  A   is 

135    degrees.     Problem  4:   0.005.      Problem  6:   The  probable 

.    ,  r    ,        1  T  •     0.00 1         ,     ,  . 

error  of  the  mean  of  the  three  readmgs  is  — -^,  and  that  ot 

the  difference   of    level   of  two  stations  is   this  multiplied  by 
V2;  then  for  the  130  stations  there  are  129  difTerences  of  level, 
and  the  probable  error  of  the  final  result  is  0.0093  feet. 

Article  107. — The  proof  of  this  method  may  be  made  in 
the  following  manner:  Let  x^  and  y,'  be  the  adjusted  values 
of  the  observations  x,   and  y^  ,  so   that  the  residual  errors  are 


§  171.  ANSWERS    TO  PROBLEMS;    AND  NOTES.  217 

r/  —  ;r,  and  r/  —J',-  Then  the  most  probable  values  of  5"  and 
7"  are  to  be  found  from  the  condition 

^/(jc/  —  xX  -\-  ^{y'l    ^  JiY  —  ^  minimum. 

The  adjusted  points  all  lie  upon  the  line  whose  equation  is 
y  ^  S-v -\- T.  Now  let  a  second  line  be  drawn  through  the 
observed  point  whose  co-ordinates  are  jt^  and/,,  and  the  ad- 
justed point  whose  co-ordinates  are  x/  and  j'/ ;  its  equatJun  h 
y  —  j\  =  S'{x  —  x\).  By  combining  this  with  the  equation  of 
the  required  line  the  values  of  the  residual  errors  are  deduced, 
whence  the  above  condition  becomes 

This  is  to  be  made  a  minimum  for  S' ,  S,  and  T  separately. 
Taking  the  derivative  with  respect  to  S'  and  equating  to  zero 
there  is  found  5'5-f-/ =  O,  which  gives  the  inclination  S' m 
terms  of  S.  Again,  differentiating  with  respect  to  vS  and  T 
there  are  deduced  two  equations  in  vS  and  T,  namely, 

S"-2.xy  -  S"  T2x  -  S2f  +  2ST'2y  ~  nST'' ^pS^x""- piixy  \ pT'2x=o, 
5Xr  +  nT-2y  =  o, 

and  the  solution  of  these  gives  values  for  5  and  T  which  agree 

with  the  results  stated  in  the  text. 

Article  112. — Problem  6:  Let  /  represent  the  population  in 
millions  and  x  the  number  of  decades  since  1800.  Then  usinp" 
the  ten  censuses  from  1790  to  1880,  there  is  found 

P  =  4-97  +  0.873.V  +  0.581^', 

which  gives  59  890  000  for  1890,  while  the  actual  enumeration 
was  62  870  000.  Again,  taking  the  seven  censuses  from  1820 
to  1880,  there  is  found 

p  =  j.2g  —  0.280A"  +  0.689JC*, 

which  gives  60579000  for  1890,  an  accordance  more  satisfac- 
tory.    The  sum  of  the  squares  of  the  residual  errors  for  the 


2l8  APPENDIX  AND    TABLES.  XI. 

latter  formula  is  1.35,  while  for  the  same  seven  census  years 
the  former  gives  3.34. 

Article  113. — Whether  observations  shall  be  independeiU 
or  conditioned  depends  in  general  upon  the  selection  of  the 
unknown  quantities  whose  values  are  to  be  determined.  Thus 
if  A  OB,  BOC,  and  AOC  are  angles  measured  at  a  station  O, 
the  observation  equations  are  independent  if  x  and  y  be  put 
for  two  of  these  angles.  But  if  x,  y,  and  z  are  taken  as  the 
three  quantities,  these  are  conditioned  by  the  necessary  rela- 
tion that  the  sum  of  two  of  them  is  equal  to  the  third. 

Article  126. — -Problem  I  :  Refer  to  problem  4  of  Article 
59.  Problem  2:  x  ^  93°  48'  I4".99.  J  =  5i°  54'  ^9" M,  -  = 
34°  16'  49",22.  Problem  5  :  This  was  the  problem  proposed 
by  Patterson  in  1808,  and  by  whose  aiscussion  Adrain  was  led 
to  the  discovery  of  the  principle  of  least  squares. 

Article  144. — The  arithmetical  mean  of  more  than  two  ob- 
servations is,  in  strictness,  the  most  probable  value  only  when 
the  results  of  the  measurements  are  unknown.  If  the  mind 
knows  the  values  of  the  measurements,  it  instinctively  assigns 
greater  reliability  to  some  than  to  others,  and  hence  the  weights 
are  not  equal.  For  example,  let  J/j  =  40,  Af^  =  5I)  ^^^3  =  5- 
be  three  observations  of  the  same  quantity :  it  is  reasonable 
to  suppose  that  ilf,  is  of  less  reliability  than  the  others,  while 
the  method  of  the  mean  assiijns  it  the  same  weight.  Theorv 
has  not  been  able  to  determine  what  theoretical  weights 
should  be  assigned  in  a  case  like  this,  but  probably  an  ap 
proach  to  them  might  be  secured  by  taking  the  reciprocal  of 
{M^  —  M^Y  +  (yJ/,  —  MJ-  as  the  weight  of  J/,,  the  reciprocal 
of  {M,  -  M)''^{M,  -  A/,y  as  the  weight  of  Jl/„  and  that  of 
{Af,  —  MJ'  +  (J/3  —  Af^f  as  the  weight  of  A/,.  For  the  above 
numerical  example  this  gives  2^^  as  the  weight  of  40,  j^j  as 
the  weight  of  51,  and  y^^  as  the  weight  of  52,  from  which 
results  the  general  mean  c  =  49.18,  whereas  the  arithmetical 
mean  is  47.67. 


§172-  •    DESCRIPTION  OF   THE    TABLES.  219 


Description  of  tJie   Tables. 

172.   Tables  I  and  II  give  values  of  the  probability  integral 

(4) ;  the  first  for  the  argument  hx,  and  the  second  for  the  argu- 

'lix  X      ^     ,      , 

ment — ,  or  -.     In  both  cases  the  arrangement  is  like  that 

0.4769        r 

of  logarithmic  tables,  and  needs  no  explanation.     The  use  of 

Table  I  is  illustrated  in  Arts.  32  and  n,  and  that  of  Table  II 

in  Art.  128.     These  tables  were  first  given  by  Encke  in  1832, 

and  were  computed  by  him  from  a  table  of  the  values  of  fc~''dt, 

which  was  published  by  Kramp  in  1799. 

Tables  III  and  IV  give  values  of  the  co-efficients  which 
occur  in  the  formulas  for  probable  error  for  values  of  //.  Table 
III  applies  to  the  usual  formulas  (20)  and  (21),  and  its  use  is 
illustrated  in  Art.  82.  Table  IV  applies  to  the  shorter  formulas 
(35)  and  (36),  and  its  use  is  illustrated  in  Art.  84.  These  tables 
were  computed  by  Wright,  and  first  published  in  "  The  Ana- 
lyst"  for  1882,  vol.  i\,  p.  74. 

Table  V  gives  four-place  logarithms  of  numbers,  and  Table 
VI  gives  four-place  squares  of  numbers.  The  latter  will  be 
found  very  useful  for  obtaining  the  squares  of  residuals.  It 
may  be  also  used  in  forming  the  co-efficients  in  normal  equa- 
tions, and  for  other  purposes.  For  instance,  the  co-efficient 
\cb]  may  be  written 

and  the  sums  [«'],  [//],  and  [{a  -f  bf'\  may  be  easily  formed  with 
the  help  of  the  table  of  squares.  This  method  has  the  advan- 
tage that  no  attention  need  be  paid  to  the  signs  of  a  and  I?, 
except  in  forming  the  sums  a  -\-  b. 

Table  VII  is  to  be  used  in  discussing  doubtful  observations 
oy  Chauvenet's  criterion,  and  its  use  is  explained  in  Art.  130. 


220 


APPENDIX  AND    TABLES. 


Table  VIII  gives  the  squares  of  reciprocals  of  numbers  from 
o.o  to  9.0,  and  may  be  used  in  the  computation  of  weights  from 
probable  errors. 


TABLE  I. 


2    /»< 

Values  of  the  Probability  Integral  -p  I  (?~'V/  for  Argument  /or  hx. 


hx. 

01234 

56789 

Diff. 

0.0 

O.I 

0.2 

0-3 
0.4 

0.0000  0.01 13  0.02260.03380.0451 

1 1 25    1236   1348    1459    1569 

---7     -335     ^443     ^55°     2657 
3286     3389     3491     3593     3694 
42S4     43S0     4475     4569     4662 

0.0564006760.07890.0901  0.1013 
1680     1790     1900    2009     21 18 
2763     2869     2974    3079    3 1  S3 
3794     3893    3992     4090     4 1 87 
4755     4847     4937     50-7     h^^l 

"3 
no 

106 

100 

92 

0.6 
0.7 
0.8 
0.9 

0.5205  0.5292  0.5379  0.5465  0.5549 
6039     61 17     6194     6270     6346 
6778     6847     6914     69S1     7047 
7421     7480     7538     7595     7651 
7969     8019     S06S     8116     8163 

0-5633  0-5716  0.5798  0.5879  0.5959 
6420     6494     6566     6638     6708 

7112     7>75     7238     7300     7361 
7707     7761     7814     7867     7918 
8209    8254     8299    S342     S385 

S3 
74 
64 

55 
45 

I.O 

I.I 

1.2 

1-3 

1.4 

0.8427  0.S468  0.8 508  0.8548  0.8586 
8802     8835    8868    8900    8931 
9103    9130    9155     9181     9205 
9340    9361     9381     9400    9419 
9523     9539    9554     9569    9583 

0.8624  0.S661  0.S698  0.8733  0.8768 
8961     8991     9020     9048     9076 
9229     9252     9275     9297     9319 
943S     9456    9473     9490    9507 
9597     961 1     9624     9637     9649 

37 

30 

-J 
18 

M 

1-5 
1.6 

1-7 
1.8 
1.9 

09661  0.9673  0.9684  0.9695  0.9706 
9763    9772     9780    9788     9796 
9S38     9844     9850    9856    9861 
9891     9895     9899    9903    9907 
992S     9931     9934    9937     9939 

0.97160.97260.97360.974509755 
9804     9811     9818     9825     9832 
9867     9872     9877     9882     9886 
99"     9915    9918    9922     9925 
9942    9944    9947     9949    995' 

10 
7 
5 
4 

t 

0 

2.0 
2.1 

2.2 

2-3 

2.4 

0-9953  0-9955  0-9957  0.9959  0.9961 
9970    9972    9973    9974    9975 
9981     9982     9983    9984    9985 
9989    9989    9990    9990    9991 
9993    9993    9994     9994    9994 

3.9963  0.9964  0.9966  0.9967  0.9969 
9976    9977     9979    99S0    9980 
9985    9986    9987     9987     9988 
9991     9992     9992    9992    9993 
9995     9995    9995    9995    9996 

2 
I 
I 

2^ 

0-9953  0.9970  0.9981  0.9989  0.9993 

0.9996  0.9998  0.9999  0.9999  0.9999 

CO 

1. 0000 

hx. 

01234 

56789 

Diff. 

VALUES  OF   THE  PROBABILITY  INTEGRAL. 


221 


TABLE    11. 


or    . 


Sl'n 

Jo                                                            0.476 

9       r 

r 

01234 

56789 

Diff. 

o.o 

0.0000  0.00 1;4  0.0108  0.016 1  0.0215 

0.0269  0.0323  0.0377  0.0430  0  0484 

54 

O.I 

0538     0591     0645     0699     0752 

0806     0859     0913     0966     1020 

54 

0.2 

1073     J '26     1 180     I2J3     1286 

•339     1392     1445     '498     155' 

53 

0-3 

,603     1656     1709     1 76 1     18 14 

1866     1918     197 1     2023     2075 

52 

;      0-4 

2127     2179     2230     2282     2334 

2385     2436     2488   .2539     2590 

5' 

'    0-5 

0.2641  0.2691  0.2742  02793  0.2843 

0.2S93  0.2944  0.2994  0.3043  0.3093 

50 

0.6 

yM    3192     3242     3291     3340 

3389     3438     3487     3535     3583 

49 

0.7 

3632     36S0     3728     3775    3823 

3S70     3918     3965     4012     4059 

46 

0.8 

4105    4152     4198    4244     4290 

4336    4381     4427     4472     45'7 

45 

0.9 

4562     4606     4651     4695     4739 

4763     4827     4860    4914     4957 

43 

I.O 

0.50000.50430  50S5  0.5128  0  5170 

0.5212  0.52540.5295  0.5337  0.5378 

4' 

I.I 

5419     5460     5500     5540     5581 

5620     5660     5700     5739     5778 

39 

I  2 

5817     5856     5894     5932     5970 

600S     6046    6083    6120    6157 

Zl 

'•3 

6194     6231     6267     6303     6339 

6375     6410    6445    6480    6515 

35 

1.4 

6550     6584     6618     6652     66S6 

6719     6753     6786    6818     6851 

J- 

'•5 

0.6S83  0.691 5  0.6947  0.6979  0.701 1 

0.7042  0.7073  0.7 104  0.7 134  0.7 165 

^? 

1.6 

7195     7225     7255     7284     7313 

7342     7371     7400    7428     7457 

28 

17 

7485     7512     7540     7567     7594 

7621     7648     7675     7701     7727 

26 

1.8 

7753     7778     7804     7829     7854 

7S79     7904     7928     7952     7976 

24 

1.9 

8000    8023     8047     8070     8093 

81 16    8138     8161     8183    8205 

22 

2,0 

0.8227  0.82480.82700.8291  0.8312 

0.S332  0.8353  0.8373  0.8394  0.8414 

19 

2.1 

S433    8453     8473     8492     85 1 1 

8^30     8549-8567     8585    8604 

18 

2  2 

8622     8639    8657     8674     8092 

87^9     8726    8742    8759    8775 

17 

23 

8792     S808     8824     8840     8855 

8870    8886    8901     8916    8930 

15 

2.4 

8945     8960    8974     8988     9002 

9016    9029    9043    9056    9069 

13 

2-5 

0.9082  09095  0.9108  0.91 21  0.9133 

0.9146  0.91 58  0.9170  0.9182  0.9193 

12 

2.6 

9205     9217     9228     9239     9250 

9261     9272     9283    9293    9304 

10 

2-7 

9314     9324     9334     9344     9354 

9364    9373    9383    9392    9401 

9 

2.8 

9410    9419     9428     9437     9446 

9454     9463     9471     9479     9487 

8 

2.9 

9495    9503    95"     95 '9    9526 

9534     9541     9548    9556    9563 

7 

3-0 

0.9570  0.9577  0.9583  0.95900.9597 

0.9603  0.9610  0.9616  0.9622  0.9629 

6 

3-1 

9635    9641     9647     9652     9658 

9664     9669     9675     9680     9686 

5 

9691     9696    9701     9706    97 1 1 

9716    9721     9726    9731     9735 

5 

9740    9744    9749    9753     9757 

9761     9766    9770    9774    9778 

4 

3-4 

9782     9786    9789    9793    9797 

9800    9804    9S07     98 1 1     9814 

4 

3- 

0.9570  0.9635  0.9691  0.9740  0.9782 

0.98 18  0.9848  0.9874  0.9896  0.991 5 

4- 

9930    9943    9954    9963    997o 

9976    9981     9985    9988    9990 

5- 

9993     9994    999^    9997     9997 

9998    9998    9999    9999    9999 

00 

1. 0000 

X 

r 

01234 

56789 

Diff. 

222 


APPENDIX  AND    TABLES. 


TABLE   III. 
For  Computing  Probable  Errors  by  Formulas  (20)  and  (21). 


n. 

0.6745 

V//  —  r 

0.6745 

n. 

0.6745 

V«  — r 

0.6745 

\JH[n-i) 

V// («-!)■ 

40 

0.1080 

O.OI7I 

41 

1066 

0167 

2 

0.6745 

0.4769 

42 

1053 

0163 

3 

4769 

2754 

43 

1 04 1 

0159 

4 

3894 

1947 

44 

1029 

0155 

5 

0.3372 

0.1508 

45 

0.1017 

0.0152 

6 

3016 

1231 

46 

1005 

0148 

7 

2754 

1041 

47 

0994 

0145 

8 

2549 

0901 

48 

0984 

0142 

9 

2385 

0795 

49 

0974 

0139 

10 

0.2248 

0.07 1 1 

50 

0.0964 

0.0136 

II 

2133 

0643 

51 

0954 

0134 

12 

2029 

0587 

52 

0944 

OI3I 

13 

1947 

0540 

53 

0935 

0128 

14 

1871 

0500 

54 

0926 

0126 

15 

0.1S03 

0.0465 

5^ 

0.0918 

0.0124 

16 

1742 

0435 

56 

0909 

0122 

17 

1686 

0409 

57 

0901 

OII9 

18 

1636 

0386 

58 

0893 

OII7 

19 

1590 

0365 

59 

08S6 

OII5 

20 

0.1547 

0.0346 

60 

0.0878 

O.OII3 

21 

1508 

0329 

61 

0871 

01  I  I 

22 

1472 

0314 

62 

0864 

Olio 

23 

1438 

0300 

63 

0857 

0108 

24 

1406 

02S7 

64 

0850 

0106 

25 

0.1377 

0.0275 

65 

0.0843 

0.0105 

26 

1349 

0265 

66 

0837 

0103 

27 

1323 

0255 

67 

0830 

OIOI 

28 

1298 

0245 

68 

0824 

0100 

29 

1275 

0237 

69 

081S 

0098 

30 

0.1252 

0.0229 

70 

0.08 1 2 

0.0097 

31 

1231 

0221 

71 

0806 

0096 

121 1 

0214 

72 

oSoo 

0094 

1 192 

0208 

73 

0795 

0093 

34 

1174 

0201 

74 

0789 

0092 

35 

0.1157 

0.0196 

75 

0.0784 

0.0091 

36 

1140 

0190 

80 

0759 

0085 

37 

II 24 

0185 

85 

0736 

0080 

38 

1109 

0180 

90 

0713 

0075 

39 

1094 

0175 

100 

i 

0678 

0068 

FOR    COMPUTING   PROBABLE  ERRORS. 


223 


TABLE  IV. 

For  Computing  Probable  Errors  by  Formulas  (35)  and  (36). 

n. 

0-8453 

0.8453 
n\^  n  —  \ 

n. 

0.8453 

0.8453 

\ln[u-x) 

40 

0.0214 

0.0034 

41 

0209 

0033 

2 

0.597S 

0.4227 

42 

0204 

0031 

^ 
0 

3451 

1993 

43 

0199 

0030 

4 

2440 

1220 

44 

0194 

0029 

5 

0. 1 S90 

0.0S45 

45 

0.0190 

0.0028 

6 

1543 

0630 

46 

0186 

0027 

7 

1304 

0493 

47 

0182 

0027 

S 

1 130 

0399 

48 

0178 

0026 

9 

0996 

033- 

49 

0174 

0025 

10 

0.0891 

0.0282 

50 

O.0171 

0.0024 

I  r 

0S06 

0243 

5' 

or  67 

0023 

12 

0736 

0212 

5- 

0164 

0023 

13 

0677 

0188 

53 

0161 

0022 

14 

0627 

0167 

54 

01 58 

0022 

1=; 

0.0583 

0.0151 

55 

0.0155 

0.0021 

16 

0546 

0136 

56 

0152 

0020 

17 

0513 

0124 

57 

0150 

0020 

18 

04S3 

01 14 

5S 

0147 

0019 

19 

0457 

0105 

59 

0145 

0019 

20 

0.0434 

0.0097 

60 

0.0142 

0.0018 

21 

0412 

0090 

61 

0140 

0018 

2  "^ 

0393 

00S4 

62 

0137 

0017 

0  -^ 

0376 

0078 

63 

0135 

0017 

-4 

0360 

0073 

64 

0133 

0017 

25 

0.034  s 

0.0069 

65 

0.0 1 3 1 

0.0016 

26 

033- 

0065 

66 

0129 

0016 

27 

0319 

0061 

67 

0127 

0016 

28 

0307 

0058 

68 

0125 

0015 

29 

0297 

0055 

69 

0123 

0015 

30 

0.02S7 

0.0052 

70 

0.0122 

0.0015 

3' 

0277 

0050 

71 

0120 

0014 

32 

026S 

0047 

72 

oiiS 

0014 

33 

0260 

0045 

73 

0117 

0014 

34 

0252 

0043 

74 

0115 

0013 

35 

0.0245 

0.0041 

75 

0.0113 

0.0013 

36 

0238 

0040 

80 

0106 

0012 

37 

0232 

0038 

85 

0100 

001 1 

3« 

0225 

0037 

90 

0095 

0010 

39 

0220 

0036 

100 

0085 

oooS 

224 


APPENDIX  AND    TABLES. 


TABLE    V. — Common  Losrarithms. 


n 

lO 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Diff 

0000 

0043 

00S6 

01  28 

0170 

0212 

0233 

0294 

0334 

0374 

42 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0C45 

06S2 

0719 

0755 

38 

12 

0792 

0S28 

0S64 

0899 

0934 

0969 

1004 

103S 

1072 

1106 

35 

13 

'1 39 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

'399 

1430 

32 

14 

1 46 1 

1492 

1523 

1553 

1584 

1614 

1044 

1(^73 

1703 

1732 

30 

15 

1761 

1790 

1S18 

1S47 

1875 

1903 

1931 

1959 

19S7 

2014 

28 

16 

2U41 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

27 

17 

2304 

2330 

2355 

23S0 

2405 

2430 

2455 

24S0 

2  5tM 

2529 

25 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

24 

19 

2783 

2810 

2833 

2S56 

2S78 

2900 

2923 

2945 

2967 

29S9 

22 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

21 

3222 

3243 

3263 

32S4 

3304 

3324 

3345 

3365 

33S5 

3404 

20 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

19 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

18 

24 

3802 

3820 

383S 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

18 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

17 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

42S1 

429S 

17 

27 

43'4 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

16 

28 

4472 

44S7 

45U2 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

15 

29 

4624 

4639 

A^zA 

4669 

4683 

4698 

4713 

472S 

4742 

4757 

15 

30 

4771 

47S6 

4S00 

4814 

4829 

4S43 

4857 

4S71 

48S6 

4900 

14 

3t 

4914 

492S 

4942 

4955 

4969 

49S3 

4997 

5011 

5024 

5038 

14 

32 

5051 

^065 

5079 

5092 

5105 

5119 

5'32 

5'45 

5159 

5172 

13 

33 

5'85 

5198 

521 1 

5224 

5237 

5250 

5263 

5276 

52S9 

5302 

13 

34 

5315 

532S 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

13 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

12 

3^ 

5563 

5575 

55S7 

5599 

.«;6ii 

5623 

5635 

5647 

5658 

56-0 

12 

37 

50S2 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

12 

38 

579S 

5809 

5S21 

5832 

5843 

5855 

5866 

5S77 

5S8S 

5899 

II 

39 

591 1 

5922 

5933 

5944 

5955 

5966 

5977 

59S8 

5999 

6010 

II 

40 

60 » I 

6031 

6042 

6053 

6064 

607  i 

6085 

6096 

6107 

6117 

1 1 

41 

612S 

613S 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

II 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

10 

43 

6335 

6345 

6355 

6365 

6375 

6385 

^395 

6405 

6415 

6425 

10 

44 

6435 

6444 

6454 

6464 

6474 

6484 

^493 

6303 

6513 

6=22 

10 

45 

6532 

6542 

6551 

6561 

6571 

65  So 

6590 

6599 

6609 

6618 

10 

46 

66?8 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

9 

47 

6721 

6730 

6735 

6749 

6758 

6767 

6776 

6785 

6794 

6S03 

9 

48 

6812 

6821 

6S30 

68:^9 

6848 

6S57 

6S66 

6S75 

6:84 

6893 

9 

49 

6902 

69 II 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

9 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

9 

51 

7076 

7084 

7093 

7101 

71 10 

7118 

7126 

7135 

7143 

7152 

8 

52 

7160 

7168 

7177 

71S5 

7193 

7202 

7210 

7218 

7226 

7235 

8 

53 

7-^43 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

8 

54 
ft 

7^2^ 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

73S8 

7396 

8 
Diff. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

COMMON  LOGARITHMS. 


225 


TABLE    V. — Common  Logarithms. 


1   " 
55 

0 

I 

2 

3 

4 

5 

6 

7__ 

8 

9 

7474 

Diflf. 
8 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

56 

7482 

7490 

7497 

7505 

75>3 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7S82 

7509 

7=97 

7604 

7612 

7619 

7627 

58 

7^)34 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

:'767 

7774 

do 

77S2 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7S46 

7 

6' 

7853 

7800 

7868 

7875 

78S2 

7889 

7896 

79U3 

7910 

79'7 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7866 

7973 

7980 

7987 

t% 

7993 

8loo 

8007 

8014 

8021 

8028 

8035 

8(->4i 

804  8 

8055 

'  ^.^ 

8062 

8069 

8075 

80S2 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8 1 62 

8169 

8176 

8182 

8189 

7 

66 

8195 

8202 

82U9 

8215 

S222 

822S 

8235 

82.41 

8248 

8254 

67 

S261 

8267 

8274 

82S0 

82S7 

8293 

8299 

8506 

8'?12 

8319 

6S 

8325 

8331 

S338 

8344 

8351 

8357 

S363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

84S2 

84S8 

8494 

8500 

8506 

6 

71 

8513 

85'9 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

S5S5 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8fe57 

8663 

8669 

8675 

868 1 

8686 

74 

8692 

S698 

8704 

8710 

S716 

8722 

8727 

8733 

8739 

87-15 

75 

875' 

S756 

8762 

8768 

8774 

8779 

S7S5 

8791 

8797 

8802 

6 

76 

S8u8 

8S14 

8820 

S825 

8S31 

8837 

8S42 

8848 

8854 

8S=;g 

77 

SS65 

8S71 

8S76 

8S82 

8SS7 

8893 

8ogg 

S904 

8910 

8915 

7S 

8921 

S927 

8932 

893S 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

QO42 

9047 

9053 

905  S 

9063 

9069 

9074 

9079 

5 

81 

9C85 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9:28 

9'33 

82 

9' 38 

0143 

9149 

9154 

9159 

9165 

9170 

9'75 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

Q2I7 

9222 

9227 

9232 

923S 

84 

9243 

9248 

9253 

925S 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

5 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

959  J 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9304 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

go 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

5 

91 

9590 

9-95 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

96^8 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

g6So 

93 

96S5 

96S9 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

973 1 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9S05 

9S09 

9814 

9818 

4 

96 

9823 

9827 

9832 

9836 

9841 

9S45 

9850 

9854 

9859 

9S63 

97 

9S68 

9872 

9877 

9881 

98S6 

9890 

9S94 

9899 

9903 

990S 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

;/ 

9956 

9961 

99^' 5 

9969 

9974 

Q978 

9983 

9987 

999 1 

gggh 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9   Diff. 

226 


APPENDIX  AND    TABLES. 


TABLE    VI. —  Squares  of  Numbers. 


n 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 
22 

I.O 

I.OCO 

J. 020 

i.04(j 

1. 061 

1.082 

1. 103 

1. 124 

1. 145 

1.166 

1.1S8 

I.I 

1. 210 

1.232 

1.254 

1.277 

1.300 

1.323 

1.346 

i-36g 

1.392 

1.416 

24 

1.2 

1.440 

1.464 

1.48.S 

1-513 

1.538 

1-563 

1.5S8 

1. 613 

1-638 

1.664 

26 

1-3 

1 .  690 

1.716 

1.742 

1-769 

1.796 

1.823 

1.850 

1-877 

1.904 

1-932 

28 

1.4 

i.9()0 

1.988 

2.016 

2.045 

2.074 

2.103 

2.132 

2. 161 

2 , 1  go 

2.220 

30 

1-5 

2.250 

2.2S0 

2.310 

2.341 

2.372 

2.403 

2.434 

2.465 

2.496 

2.528 

32 

1.6 

2.560 

2.592 

2.624 

2.657 

2.690 

2.723 

2-756 

2-789 

2.822 

2.856 

34 

1-7 

2.8go 

2.924 

2.g58 

2.993 

3.028 

3  063 

3.09S 

3-133 

3-168 

3.204 

36 

i.S 

3.240 

3.276 

3.312 

3-349 

3.386 

3.423 

3.460 

3-497 

3.534 

3-572 

38 

1. 9 

3.610 

3.648 

3.686 

3-725 

3.764 

3.803 

3.842 

3-8S1 

3.920 

3-960 

40 

1       2.0 

4.OCO 

4.040 

4.080 

4. 121 

4.162 

4.203 

4  244 

4.285 

4.326 

4.368 

42 

2.1 

4.410 

4.452 

4-494 

4.537 

4-580 

4.623 

4.666 

4.709 

4-752 

4-796 

44 

2.2 

4.S40 

4.8S4 

4.92S 

4.973 

5.01S 

5.063 

5-ioS 

5.153 

5.198 

5-244 

46 

2.3 

5  290 

5.336 

5-382 

5-429 

5.476 

5.523 

5-570 

5-617 

5.664 

5.712 

48 

2.4 

5.760 

5.808 

5.855 

5.905 

5.954 

6.003 

6.052 

6.101 

6.150 

6  200 

50 

2-5 

6.250 

6.300 

6.350 

6.401 

6.452 

6.503 

6.554 

6.605 

6.656 

6. 70S 

52 

2  6 

6.760 

6.812 

6.864 

6-917 

6.970 

7.023 

7.076 

7.129 

7.182 

7-236 

54 

2.7 

7.290 

7.344 

7.398 

7-453 

7.508 

7.563 

7.61S 

7-673 

7.728 

7-784 

56 

2.8 

7.840 

7896 

7.952 

8.  cog 

8.066 

8.123 

8.180 

8.237 

8.294 

8.352 

■:8 

2.9 

S.410 

S.468 

8.526 

8.585 

8.644 

8.703 

8.762 

8.821 

8.880 

8.940 

60 

3.0 

g.ooo 

9.060 

9.120 

9.181 

9.242 

9.303 

9.364 

9425 

9.486 

9-548 

62 

3.1 

9  610 

9.672 

9  734 

9.797 

9.860 

9  923 

9.gS6 

10.05 

10.11 

10.18 

6 

3-2 

10.24 

10.30 

10.37 

10.43 

10.50 

10.56 

10.63 

10.69 

10.76 

10.82 

7 

3  3 

10.89 

10.96 

11.02 

II. og 

II. 16 

11.22 

11.29 

11.36 

1 1.42 

11.49 

7 

3-4 

11.56 

11.63 

11.70 

11.76 

1 1. S3 

1 1. go 

11.97 

12.04 

12. II 

12. iS 

7 

3-5 

12.25 

12.32 

12.39 

12.46 

12.53 

12.60 

12.67 

12.74 

12  82 

12. Sg 

7 

3-6 

12.90 

13.03 

13.10 

13.18 

13  25 

13  32 

13.40 

13-47 

13  54 

14.62 

7 

3-7 

13.69 

13.76 

13.84 

13-91 

1399 

14  06 

14.14 

14  21 

14.29 

14.36 

8 

3-S 

14.44 

14.52 

14.59 

14-67 

14.75 

14.82 

14-90 

14  98 

m.05 

15.13 

8 

3.9 

15.21 

15.29 

15.37 

15.44 

15.52 

15.60 

15.68 

15.76 

15.84 

15.92 

8 

4.0 

16.00 

16.08 

16.16 

16.24 

16.32 

,'6  40 

16.48 

16.56 

16.65 

16.73 

8 

4   ' 

16  Si 

16.89 

1 6. 07 

17.06 

17.14 

17  22 

17  31 

17.39 

17.47 

17.56 

8 

42 

17.64 

17.72 

17.81 

17.89 

17.98 

18  06 

18.15 

18.23 

18.32 

18.40 

9 

4  3 

18.49 

\%.^% 

18  66 

18  75 

18.84 

18. g2 

19.01 

19. 10 

19  18 

ig  27 

9 

4.1 

19.36 

19.45 

19.54 

19.62 

19-71 

19  80 

ig.89 

19.98 

20.07 

20.16 

9 

4  5 

20.25 

20.34 

20.43 

20.  =  2 

20.61 

20.70 

20.79 

20.88 

20.  gS 

21.07 

9 

46 

21.16 

21.25 

21.34 

21.44 

21.53 

21.62 

21.72 

21.81 

21. gf) 

22  00 

9 

4.7 

22. og 

22.18 

22  28 

22.37 

22.47 

22.56 

22.66 

22.75 

22.85 

22  g4 

10 

4.8 

23.04 

23.14 

23.23 

23-33 

23-43 

23.52 

23.62 

23.72 

23.81 

23.91 

10 

4-9 

24  01 

24.11 

24.21 

24.30 

24  40 

24.50 

24  60 

24-70 

24  80 

24.<;0 

10 

?.o 

25.00 

25  10 

25.20 

25.30 

25.40 

25.50 

25.60 

25  70 

25.81 

25.91 

10 

5-1 

26.01 

26.11 

26.21 

26.32 

26.42 

26  52 

26.63 

26.73 

26.83 

26.94 

10 

5  2 

27.04 

27.14 

27.25 

27.35 

27.46 

27.56 

27.67 

27-77 

27.88 

27.98 

II 

5-3 

28.09 

28.20 

28.30 

28.41 

28.52 

28.62 

28.73 

28. S4 

2S.94 

29-05 

ri 

5-4 

.1 

29.16 

29.27 

29  38 

29.48 

29.59 

20.70 

2g.8i 

29. g2 

30.03 

30.14 

^'     i 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Diff.  i 

SQUARES    OF  NUMBERS. 


227 


TABLE    VI.— Squares  of  Numbers. 


n 

0 

1           2 

3 

4 

5 

6 

7 

8 

9 

Diff. 
11 

5-5 

30  25 

30.36  30.47 

30.58 

30.69 

30.  So 

30.91 

31.02 

31-14 

31-25 

5 

6 

31.36 

31.47  31.58 

31.70 

31.81 

31.92 

32.04 

32.15 

32.26 

32-38 

II 

5 

7 

32.49 

32.60  32.72 

32.83 

32-95 

33.06 

33-18 

33  29 

33-41 

33.52 

12 

5 

S 

33.64 

33-76  33.87 

33-99 

34.11 

34-22 

34.34 

34-46 

34-57 

34.60 

12 

5 

9 

34.S1 

34-93  3505 

35-16 

35-2S 

35-40 

35.52 

35-64 

35-76 

35-88 

12 

6 

0 

36.00 

36.12   36.24 

36.36 

36.48 

36.60 

3672 

36  S4 

36  97 

37-00 

12 

6 

I 

37.21 

37.33  37.45 

37-58 

37-70 

37-82 

37-95 

38.07 

38.19 

38-32 

12 

6 

2 

38.44 

38. 56  38.69  38.81 

38.94 

39.06 

39.19 

39-31 

39-44 

39-56 

13 

6 

3 

3969 

39  82  39.94 

40.07 

40.20 

40  32 

40.45 

40.5S 

40.70 

40.53 

13 

6 

4 

40.96 

41.09  41.22 

41-34 

41.47 

41.00 

41.73 

41. S6 

41.99 

42.12 

13 

6 

5 

42.25 

42.38  42. m 

42.64 

42.77 

42.90 

43-03 

,13.16 

43-30 

43-43 

13 

6 

6 

43-56 

4369  43.82 

4396 

44.09 

44.22 

44  36 

44  49 

44.62 

44.76 

13 

-6 

7 

44.89 

45.02  45.16 

45  29 

45-43 

45.56 

45  70 

45.83 

45-97 

46.10 

14 

6 

8 

46.24 

46.38  46.51 

46.65 

46.79 

46.02 

47.06 

47.20 

47-33 

-17-47 

14 

6 

9 

47.61 

47.75  47-89 

48.02 

48.16 

48.30 

4S.44 

4S-5S 

48.72 

4S.86 

14 

7 

0 

49.00 

49.14  49.28 

49.42 

49-56 

49  70 

49.84 

49-98 

50.13 

50-27 

14 

7 

I 

50.41 

50.55  50.69 

50.84 

50-98 

51.12 

51.27 

51-41 

51-55 

51.70 

14 

7 

2 

51.84 

51.98  52.13 

52.27 

52.42 

52.56 

52.71 

52  85 

53.00 

53-14 

15 

7 

3 

53.29 

53-44  53.58 

53-73 

53-88 

54.C2 

54-17 

54-32 

54-46 

54-61 

15 

7 

4 

54-76 

54.91   55.06 

55-20 

55-35 

55.50 

55-65 

55. So 

55-95 

56.10 

15 

7 

5 

56.25 

56.40  56.55 

56.70 

56.85 

57-00 

57-15 

57.30 

57-46 

57.61 

15 

7 

6 

57-76 

57.91   58. 06 

58. 22 

58.37 

58  52 

5S.68 

58  83 

58.98 

59-14 

15 

7 

7 

59-29 

59.44  59.60 

59-75 

59.91 

60.06 

60.22 

60.37 

60.53 

60.68 

16 

7 

8 

60.84 

61.00  61.15 

61.31 

61.47 

61.62 

61.78 

61.94 

62.09 

62.25 

16 

7 

9 

62.41 

62.57  62.73 

62. 88 

63.04 

63  .CO 

63-36 

63.52 

63.68 

6384 

16 

8 

0 

64.00 

64.16  64.32 

64.48  64.64 

64.80 

64  96 

6^.12 

65.29 

65-45 

16 

8 

I 

65.61 

65.77  65.93 

66.10 

66.26 

66.42 

66.59 

66  7:; 

66.91 

67.08 

16 

8 

2 

67.24 

67.40  67.57 

67-73 

67.90 

68.06 

68.23 

68.39 

68  56 

68. 72 

17 

8 

3 

68.89  69.06  69.22 

6939 

69  56 

69.72 

69.89 

70.06 

70.22 

70.39 

:  -7 

8 

4 

70.56 

70.73  70.90 

71.06 

71.23 

71.40 

71.57 

71.74 

71.91 

72.08 

17 

8 

5 

72.25 

72.42  72.59 

72.76 

72.93 

73. TO 

73-27 

73-44 

73-62 

73.79 

17 

8 

6 

73-96 

74.13  74.30 

74.48  74.65 

74.82 

75. CO 

75-17 

75  34 

75.52 

17 

8 

7 

7569 

75.86  76.04 

76  21 

76.39 

76.R6 

76.74 

76.91 

77-09 

77.26 

18 

8 

8 

77-44 

77.62   77.79 

77.97 

78.15 

78-32 

78. so 

78.68 

7885 

79-03 

18 

8 

9 

79.21 

79-39  79-57 

79-74 

79.92 

80.10 

So.  28 

So.  46  80.64 

So  82 

18 

9 

0 

Si.oo 

81.18  81.36 

81.54 

81.72 

81.90 

82. 08 

82. 26 

82.45 

82.63 

18 

9 

i 

82.81 

82.99  83.17 

83.36 

83.54 

83-72 

8-;. 91 

84.09 

84.27 

84.46 

18 

9 

2 

S4.64  84.8"  85.01 

85.19 

85-38 

85-56 

85.75 

85-93 

86.12 

86  30 

19 

9 

3 

86.49 

86.68  S6.86 

8705 

87.24 

87-42 

87.61 

87.80 

87.9S 

88.17 

19 

9 

4 

88.36 

88.55  88.74 

88.92 

89.11 

89,30 

89.49 

89.68 

89,87 

90.06 

19 

9 

5 

90.25 

00.44  90.63 

90.82 

gi.oi 

91.20 

91-39 

91. 58 

91-78 

91.97 

19 

9 

.6 

92.16 

92.35  92.54 

92.74 

92.93 

93.12 

93-32 

9^^-51 

93.70 

93-90 

19 

9 

7 

94.09 

94.28  94.48 

94-67 

94.87 

95.06 

95.26 

95-45 

9565 

95-84 

20 

9 

8 

96  04 

96.24  96.43 

96.63  96.83 

97  02 

97.22 

97  42 

97.61 

97.81 

20 

9 

9 

98.01 

98.21  98.41 

98.60 

98.80 

9().oo 

5 

99.20 
6 

99.40 

7 

99.60 
8 

99.80 
9 

20 
Diff. 

1       " 

0 

I           2 

3 

4 

228 


APPENDIX  AND    TABLES. 


TABLE   VII.  —  For  Applying  Chauvenet's  Criterion. 


Jl. 

t. 

;/. 

/. 

n. 

t. 

3 

2.05 

13 

3-07 

23 

341 

4 

2.27 

14 

3-'; 

24 

3-43 

5 

2.44 

15 

3.16 

25 

3-45 

6 

2-57 

16 

3-19 

30 

3-55 

7 

2.67 

17 

3.22 

40 

0-70 

8 

2.76 

18 

3.26 

50 

3.82 

9 

2.84 

19 

3-29 

75 

4.02 

10 

2  9( 

20 

3-y- 

too 

4.16 

II 

2.96 

21 

J-J3 

200 

4.48 

12 

3.02 

22 

3-3^ 

500 

4.90 

TABLE    VIII.  — Squares  of  Reciprocals. 


I 

I 

I 

;/. 

2 

;/. 

2 

;/. 

2 

ir 

ir 

>r 

0.0 

00 

25 

0. 1 600 

5-0 

0.0400 

0.1 

100.000 

2.6 

0.1479 

5-1 

0.0384 

0.2 

25.000 

2-7 

0.1372 

5-2 

0.0370 

0-3 

I  I.I  1 1 

28 

0.1276 

5-3 

0.0356 

0.4 

6.250 

2.9 

O.I  189 

5-4 

0.0343 

0.5 

4  000 

3-0 

O.IIII 

5-5 

0.0331 

0.6 

2.778 

3' 

0.1 04 1 

56 

0.0319 

0.7 

2.041 

3-2 

0.0977 

H 

0.0308 

0.8 

1.562 

3-3 

0.0918 

5-8 

0.0297 

0.9 

1-235 

3-4 

0.0865 

5-9 

0.02S7 

I.O 

1 .000 

3-5 

0.0816 

6.0 

0.0278 

I.I 

0.S264   1 

3-6 

0.0772 

6.1 

0.0269 

1.2 

0.6944   1 

7>-l 

0.0730 

6.2 

0.0260 

1-3 

0.5917 

3-8 

0.0693 

6-3 

0.0252 

1.4 

0.5102 

3-9 

0.0657 

6.4 

0.0244 

1-5 

0.4444 

4.0 

0.0625 

6.5 

0.0237 

16 

0.3906 

4.1 

0.0595 

6.6 

0.0230 

1-7 

03460 

4.2 

0.0567 

6.7 

0.0223 

1.8 

0.3086 

4-3 

0.0541 

6.8 

0.0216 

1.9 

0.2770 

4.4 

0.0517 

6.9 

0.0210 

2.0 

0. 2  500 

4-5 

0.0494 

7.0 

0.0204 

2.1 

0.226S 

4.6 

0.0473 

^5 

0.0178 

2.2 

0.2066 

4-7 

0.0453 

8.0 

0.0156 

2-3 

0.1890 

4.8 

0.0434 

8.5 

0.0138 

2.4 

0.1736 

4.9 

0.0416 

9.0 

0.0123 

INDEX. 


229 


INDEX. 


Accidental  errors,  4 

Adjustment,  i,   36,  51,   SS,   loi,   109, 
141    1S7 

Angle  measurements,  104,  171 
repetition,  106 

Angles,  3,  90,  98,  122,  163 

at  a  station,  117,  145 

in  a  quadrilateral,  147,  150 

in  a  triangle,  142 

Areas.  3,  106 

Arithmetical  mean.  42,  70,  211,  218 

Axioms,  13 

Base  lines,  100,  102 
Binomial  formula,  10 
Borings,  140 

Certainty,  6 

Chaining,  103 

Chauvenet's  criterion,  166,  228 

Coins,  throwing  of,  g 

Comparison  of  observations,  i,  66 

Conditioned   observations,  2,  57,  86, 

141,  192 
Constant  errors,  3,  169 
Constants,  214 
Correlatives,  60 
Criterion  for  rejection,  166 
Curve  of  probability,  15,  25,  204 

Declination,  magnetic,  134 
Direct  observations,  2,  41,  88 
Doubtful  observations,  166 


Earth,  temperature  of,  140 
Empirical  constants,  124 
formulas,  130 
Equal  weights,  88 
Equations,  non-linear,  200 

normal,  46,  56,  175 
observation,  58 
solution  of,  175 
Error,  definition  of,  5 

law  of,  13,  17,  22 
probability  of,  13,  162 
propagation  of,  75 
Experience,  axioms  from,  13 

Functions  of  observations,  90 

Gauss's     discussions,    22,     175,    181, 

212 
General  mean,  42,  72 
Geodesy,   151,  214 
Guessing,  problem  on,  174 

Hagen's  proof.  17,  168 
History  of  Least  Squares,  211 
Huge  error,  205 

Impossibility,  6 

Independent  observations,  2,  51,  79, 

100 
Indirect  observations,  2,  43 
Instrumental  errors,  4 

Level  lines,  44,  no,  157 
Levelling,  154 


230 


INDEX. 


Linear  measurements,  loi 
Liteiature  of  Least  Squares,  213 
Logarithmic  computation,  190 
Logarithms,  219,  224 

Magnetic  declination,  134 

Mean  error,  204 

Measure  of  precision,  34,  68 

Median,  208 

Mistakes,  4,  169 

Most  probable  value,  2,  g,  38 

Non-linear  equations,  200 
Normal  equations,  46,  56,  175 

Observation  equations,  58 

Observations,   adjustment  of,  i,  36, 
88,  loi,  109,  141 
classification  of,  2 
discussion  of,  162 
errors  of,  3,  5,  13 
precision  of,  66 
rejection  of,  166 
weights  of,  36 

Orbit  of  a  planet,  129 

Peirce's  criterion,  169 

Pendulum,  124 

Population  of  United  States,  202,  217 

Principle  of  Least  Squares,  38,  211 

Probability,  6,  9. 

Probability  curve,  13,  25,  68 

integral,  27,  220,  221 
of  error,  13,  162,  212 

Probable  error,  66,  70,  72,  79,  86,  92, 
195,  204 

Propagation  of  error,  75 


Quadrilateral,  147 
Quetelet's  statistics,  175 

Reciprocals,  squares  of,  228 
Rejection  of  observations,  166 
Repetition  of  angles,  106 
Residual,  5,  39 
Rivers,  velocity  in,  131 

Shooting  at  target,  13,  165 
Social  statistics,  172 
Solution  of  equations,  56,  175 
Squares  of  numbers,  227 

reciprocals,  228 
Station  adjustment,  118,  145 
Statistics,  162,  172 

Tables,  220-228 
Target  shots,  13,  165,  170 
Theory  and  experience,  31 
Triangle  adjustment,  59,  142 
Triangulation,  152 

Uncertainty  of  median,  210 

probable  error,  206 
Unequal  weights,  51,  95,  122 

Velocity  observations,  131,  138 

Weighted  mean,  43 

observations,  37,  51,  187 

residuals,  39 
Weights,  36,  69,  196 
Wright's  probatle-error  tables,  219 
222,  223 


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